1. Notation and Algebra of Functions

2. Notation and Algebra of Functions
Functions are procedures that assign a unique output to each
(valid) input.

3. Notation and Algebra of Functions
Functions are procedures that assign a unique output to each
(valid) input. Most mathematics functions are given in
mathematics formulas such as f(x) = x2 – 2x + 3 = y-the output.

4. Notation and Algebra of Functions
Functions are procedures that assign a unique output to each
(valid) input. Most mathematics functions are given in
mathematics formulas such as f(x) = x2 – 2x + 3 = y-the output.
name of the input box the defining
the function formula

5. Notation and Algebra of Functions
Functions are procedures that assign a unique output to each
(valid) input. Most mathematics functions are given in
mathematics formulas such as f(x) = x2 – 2x + 3 = y-the output.
The input box “( )” holds the input to be evaluated by the
formula.

6. Notation and Algebra of Functions
Functions are procedures that assign a unique output to each
(valid) input. Most mathematics functions are given in
mathematics formulas such as f(x) = x2 – 2x + 3 = y-the output.
The input box “( )” holds the input to be evaluated by the
formula. Hence f(2) means to replace x by the input (2) in the
formula, so f(2) = (2)2 – 2(2) + 3 = 3 = y.

7. Notation and Algebra of Functions
Functions are procedures that assign a unique output to each
(valid) input. Most mathematics functions are given in
mathematics formulas such as f(x) = x2 – 2x + 3 = y-the output.
The input box “( )” holds the input to be evaluated by the
formula. Hence f(2) means to replace x by the input (2) in the
formula, so f(2) = (2)2 – 2(2) + 3 = 3 = y.
Expressions may be formed using the above function
notation.

8. Notation and Algebra of Functions
Functions are procedures that assign a unique output to each
(valid) input. Most mathematics functions are given in
mathematics formulas such as f(x) = x2 – 2x + 3 = y-the output.
The input box “( )” holds the input to be evaluated by the
formula. Hence f(2) means to replace x by the input (2) in the
formula, so f(2) = (2)2 – 2(2) + 3 = 3 = y.
Expressions may be formed using the above function
notation. Once we know that f(2) is 3, expressions such as
2*f(2) or [f(2)]2 have definitive answers.

9. Notation and Algebra of Functions
Functions are procedures that assign a unique output to each
(valid) input. Most mathematics functions are given in
mathematics formulas such as f(x) = x2 – 2x + 3 = y-the output.
The input box “( )” holds the input to be evaluated by the
formula. Hence f(2) means to replace x by the input (2) in the
formula, so f(2) = (2)2 – 2(2) + 3 = 3 = y.
Expressions may be formed using the above function
notation. Once we know that f(2) is 3, expressions such as
2*f(2) or [f(2)]2 have definitive answers.
Example A. Let f(x) = –3x + 2 and g(x) = –2x2 – 3x + 1.
a. Evaluate f(3).

10. Notation and Algebra of Functions
Functions are procedures that assign a unique output to each
(valid) input. Most mathematics functions are given in
mathematics formulas such as f(x) = x2 – 2x + 3 = y-the output.
The input box “( )” holds the input to be evaluated by the
formula. Hence f(2) means to replace x by the input (2) in the
formula, so f(2) = (2)2 – 2(2) + 3 = 3 = y.
Expressions may be formed using the above function
notation. Once we know that f(2) is 3, expressions such as
2*f(2) or [f(2)]2 have definitive answers.
Example A. Let f(x) = –3x + 2 and g(x) = –2x2 – 3x + 1.
a. Evaluate f(3).
f(3) = –3(3) + 2

11. Notation and Algebra of Functions
Functions are procedures that assign a unique output to each
(valid) input. Most mathematics functions are given in
mathematics formulas such as f(x) = x2 – 2x + 3 = y-the output.
The input box “( )” holds the input to be evaluated by the
formula. Hence f(2) means to replace x by the input (2) in the
formula, so f(2) = (2)2 – 2(2) + 3 = 3 = y.
Expressions may be formed using the above function
notation. Once we know that f(2) is 3, expressions such as
2*f(2) or [f(2)]2 have definitive answers.
Example A. Let f(x) = –3x + 2 and g(x) = –2x2 – 3x + 1.
a. Evaluate f(3).
f(3) = –3(3) + 2 so
f(3) = –7
b. Evaluate g(–2).
g(–2) = –2(–2)2 – 3(–2) + 1
= –8 + 6 + 1 = –1.

12. Notation and Algebra of Functions
Functions are procedures that assign a unique output to each
(valid) input. Most mathematics functions are given in
mathematics formulas such as f(x) = x2 – 2x + 3 = y-the output.
The input box “( )” holds the input to be evaluated by the
formula. Hence f(2) means to replace x by the input (2) in the
formula, so f(2) = (2)2 – 2(2) + 3 = 3 = y.
Expressions may be formed using the above function
notation. Once we know that f(2) is 3, expressions such as
2*f(2) or [f(2)]2 have definitive answers.
Example A. Let f(x) = –3x + 2 and g(x) = –2x2 – 3x + 1.
a. Evaluate f(3).
f(3) = –3(3) + 2 so
f(3) = –7
b. Evaluate g(–2).

13. Notation and Algebra of Functions
Functions are procedures that assign a unique output to each
(valid) input. Most mathematics functions are given in
mathematics formulas such as f(x) = x2 – 2x + 3 = y-the output.
The input box “( )” holds the input to be evaluated by the
formula. Hence f(2) means to replace x by the input (2) in the
formula, so f(2) = (2)2 – 2(2) + 3 = 3 = y.
Expressions may be formed using the above function
notation. Once we know that f(2) is 3, expressions such as
2*f(2) or [f(2)]2 have definitive answers.
Example A. Let f(x) = –3x + 2 and g(x) = –2x2 – 3x + 1.
a. Evaluate f(3).
f(3) = –3(3) + 2 so
f(3) = –7
b. Evaluate g(–2).
g(–2) = –2(–2)2 – 3(–2) + 1

14. Notation and Algebra of Functions
Functions are procedures that assign a unique output to each
(valid) input. Most mathematics functions are given in
mathematics formulas such as f(x) = x2 – 2x + 3 = y-the output.
The input box “( )” holds the input to be evaluated by the
formula. Hence f(2) means to replace x by the input (2) in the
formula, so f(2) = (2)2 – 2(2) + 3 = 3 = y.
Expressions may be formed using the above function
notation. Once we know that f(2) is 3, expressions such as
2*f(2) or [f(2)]2 have definitive answers.
Example A. Let f(x) = –3x + 2 and g(x) = –2x2 – 3x + 1.
a. Evaluate f(3).
f(3) = –3(3) + 2 so
f(3) = –7
b. Evaluate g(–2).
g(–2) = –2(–2)2 – 3(–2) + 1
= –8 + 6 + 1 = –1.

15. Notation and Algebra of Functions
f(x) = –3x + 2 , g(x) = –2x2 – 3x + 1
c. Evaluate 2*f(3) – 4*g(–2).

16. Notation and Algebra of Functions
f(x) = –3x + 2 , g(x) = –2x2 – 3x + 1
c. Evaluate 2*f(3) – 4*g(–2).
Using the outputs from above that f(3) = –7 and g(–2) = –1
2*f(3) – 4*g(–2)
=

17. Notation and Algebra of Functions
f(x) = –3x + 2 , g(x) = –2x2 – 3x + 1
c. Evaluate 2*f(3) – 4*g(–2).
Using the outputs from above that f(3) = –7 and g(–2) = –1
2*f(3) – 4*g(–2)
= 2*(–7) – 4*(–1)

18. Notation and Algebra of Functions
f(x) = –3x + 2 , g(x) = –2x2 – 3x + 1
c. Evaluate 2*f(3) – 4*g(–2).
Using the outputs from above that f(3) = –7 and g(–2) = –1
2*f(3) – 4*g(–2)
= 2*(–7) – 4*(–1) = –10.

19. Notation and Algebra of Functions
f(x) = –3x + 2 , g(x) = –2x2 – 3x + 1
c. Evaluate 2*f(3) – 4*g(–2).
Using the outputs from above that f(3) = –7 and g(–2) = –1
2*f(3) – 4*g(–2)
= 2*(–7) – 4*(–1) = –10.
We may use the output of one function as the input for another
function.

20. Notation and Algebra of Functions
f(x) = –3x + 2 , g(x) = –2x2 – 3x + 1
c. Evaluate 2*f(3) – 4*g(–2).
Using the outputs from above that f(3) = –7 and g(–2) = –1
2*f(3) – 4*g(–2)
= 2*(–7) – 4*(–1) = –10.
We may use the output of one function as the input for another
function.
d. Evaluate f(g(0)).

21. Notation and Algebra of Functions
f(x) = –3x + 2 , g(x) = –2x2 – 3x + 1
c. Evaluate 2*f(3) – 4*g(–2).
Using the outputs from above that f(3) = –7 and g(–2) = –1
2*f(3) – 4*g(–2)
= 2*(–7) – 4*(–1) = –10.
We may use the output of one function as the input for another
function.
d. Evaluate f(g(0)).
The expression f(g(0)) means g(0) is to be the input for f(x).

22. Notation and Algebra of Functions
f(x) = –3x + 2 , g(x) = –2x2 – 3x + 1
c. Evaluate 2*f(3) – 4*g(–2).
Using the outputs from above that f(3) = –7 and g(–2) = –1
2*f(3) – 4*g(–2)
= 2*(–7) – 4*(–1) = –10.
We may use the output of one function as the input for another
function.
d. Evaluate f(g(0)).
The expression f(g(0)) means g(0) is to be the input for f(x).
Since g(0) = –2(0)2 – 3(0) + 1 = 1,

23. Notation and Algebra of Functions
f(x) = –3x + 2 , g(x) = –2x2 – 3x + 1
c. Evaluate 2*f(3) – 4*g(–2).
Using the outputs from above that f(3) = –7 and g(–2) = –1
2*f(3) – 4*g(–2)
= 2*(–7) – 4*(–1) = –10.
We may use the output of one function as the input for another
function.
d. Evaluate f(g(0)).
The expression f(g(0)) means g(0) is to be the input for f(x).
Since g(0) = –2(0)2 – 3(0) + 1 = 1, we have that
f(g(0))
= f(1)

24. Notation and Algebra of Functions
f(x) = –3x + 2 , g(x) = –2x2 – 3x + 1
c. Evaluate 2*f(3) – 4*g(–2).
Using the outputs from above that f(3) = –7 and g(–2) = –1
2*f(3) – 4*g(–2)
= 2*(–7) – 4*(–1) = –10.
We may use the output of one function as the input for another
function.
d. Evaluate f(g(0)).
The expression f(g(0)) means g(0) is to be the input for f(x).
Since g(0) = –2(0)2 – 3(0) + 1 = 1, we have that
f(g(0))
= f(1) = –3(1) + 2 = –1

25. Notation and Algebra of Functions
f(x) = –3x + 2 , g(x) = –2x2 – 3x + 1
c. Evaluate 2*f(3) – 4*g(–2).
Using the outputs from above that f(3) = –7 and g(–2) = –1
2*f(3) – 4*g(–2)
= 2*(–7) – 4*(–1) = –10.
We may use the output of one function as the input for another
function.
d. Evaluate f(g(0)).
The expression f(g(0)) means g(0) is to be the input for f(x).
Since g(0) = –2(0)2 – 3(0) + 1 = 1, we have that
f(g(0))
= f(1) = –3(1) + 2 = –1
e. Evaluate g(f(0)).

26. Notation and Algebra of Functions
f(x) = –3x + 2 , g(x) = –2x2 – 3x + 1
c. Evaluate 2*f(3) – 4*g(–2).
Using the outputs from above that f(3) = –7 and g(–2) = –1
2*f(3) – 4*g(–2)
= 2*(–7) – 4*(–1) = –10.
We may use the output of one function as the input for another
function.
d. Evaluate f(g(0)).
The expression f(g(0)) means g(0) is to be the input for f(x).
Since g(0) = –2(0)2 – 3(0) + 1 = 1, we have that
f(g(0))
= f(1) = –3(1) + 2 = –1
e. Evaluate g(f(0)).
The expression g(f(0)) means f(0) is to be the input for g(x).

27. Notation and Algebra of Functions
f(x) = –3x + 2 , g(x) = –2x2 – 3x + 1
c. Evaluate 2*f(3) – 4*g(–2).
Using the outputs from above that f(3) = –7 and g(–2) = –1
2*f(3) – 4*g(–2)
= 2*(–7) – 4*(–1) = –10.
We may use the output of one function as the input for another
function.
d. Evaluate f(g(0)).
The expression f(g(0)) means g(0) is to be the input for f(x).
Since g(0) = –2(0)2 – 3(0) + 1 = 1, we have that
f(g(0))
= f(1) = –3(1) + 2 = –1
e. Evaluate g(f(0)).
The expression g(f(0)) means f(0) is to be the input for g(x).
Since f(0) = –3(0) + 2 = 2,

28. Notation and Algebra of Functions
f(x) = –3x + 2 , g(x) = –2x2 – 3x + 1
c. Evaluate 2*f(3) – 4*g(–2).
Using the outputs from above that f(3) = –7 and g(–2) = –1
2*f(3) – 4*g(–2)
= 2*(–7) – 4*(–1) = –10.
We may use the output of one function as the input for another
function.
d. Evaluate f(g(0)).
The expression f(g(0)) means g(0) is to be the input for f(x).
Since g(0) = –2(0)2 – 3(0) + 1 = 1, we have that
f(g(0))
= f(1) = –3(1) + 2 = –1
e. Evaluate g(f(0)).
The expression g(f(0)) means f(0) is to be the input for g(x).
Since f(0) = –3(0) + 2 = 2, we have that
g(f(0)) = g(2)

29. Notation and Algebra of Functions
f(x) = –3x + 2 , g(x) = –2x2 – 3x + 1
c. Evaluate 2*f(3) – 4*g(–2).
Using the outputs from above that f(3) = –7 and g(–2) = –1
2*f(3) – 4*g(–2)
= 2*(–7) – 4*(–1) = –10.
We may use the output of one function as the input for another
function.
d. Evaluate f(g(0)).
The expression f(g(0)) means g(0) is to be the input for f(x).
Since g(0) = –2(0)2 – 3(0) + 1 = 1, we have that
f(g(0))
= f(1) = –3(1) + 2 = –1
e. Evaluate g(f(0)).
The expression g(f(0)) means f(0) is to be the input for g(x).
Since f(0) = –3(0) + 2 = 2, we have that
g(f(0)) = g(2) = –2(2)2 – 3(2) + 1 = –13

30. Notation and Algebra of Functions
The above algebra which utilizes the function notation enables
us to keep track of quantitative information efficiently.

31. Notation and Algebra of Functions
The above algebra which utilizes the function notation enables
us to keep track of quantitative information efficiently.
Example B.
a. Apples cost $8.00/box, what is the cost of x boxes of apples?

32. Notation and Algebra of Functions
The above algebra which utilizes the function notation enables
us to keep track of quantitative information efficiently.
Example B.
a. Apples cost $8.00/box, what is the cost of x boxes of apples?
At the $8.00/box, x boxes of apples cost 8x.

33. Notation and Algebra of Functions
The above algebra which utilizes the function notation enables
us to keep track of quantitative information efficiently.
Example B.
a. Apples cost $8.00/box, what is the cost of x boxes of apples?
At the $8.00/box, x boxes of apples cost 8x.
b. For an online order of boxes of apples there is a flat shipping
fee of $12. Let A(x) = the total cost of an online order of x boxes
of apples, i.e. the cost of the apples plus the $12 shipping fee,
what is A(x)?

34. Notation and Algebra of Functions
The above algebra which utilizes the function notation enables
us to keep track of quantitative information efficiently.
Example B.
a. Apples cost $8.00/box, what is the cost of x boxes of apples?
At the $8.00/box, x boxes of apples cost 8x.
b. For an online order of boxes of apples there is a flat shipping
fee of $12. Let A(x) = the total cost of an online order of x boxes
of apples, i.e. the cost of the apples plus the $12 shipping fee,
what is A(x)?
The total cost function is the cost of the apples, i.e. 8x, plus the
flat shipping fee of $12,

35. Notation and Algebra of Functions
The above algebra which utilizes the function notation enables
us to keep track of quantitative information efficiently.
Example B.
a. Apples cost $8.00/box, what is the cost of x boxes of apples?
At the $8.00/box, x boxes of apples cost 8x.
b. For an online order of boxes of apples there is a flat shipping
fee of $12. Let A(x) = the total cost of an online order of x boxes
of apples, i.e. the cost of the apples plus the $12 shipping fee,
what is A(x)?
The total cost function is the cost of the apples, i.e. 8x, plus the
flat shipping fee of $12, so that A(x) = 8x + 12.

36. Notation and Algebra of Functions
The above algebra which utilizes the function notation enables
us to keep track of quantitative information efficiently.
Example B.
a. Apples cost $8.00/box, what is the cost of x boxes of apples?
At the $8.00/box, x boxes of apples cost 8x.
b. For an online order of boxes of apples there is a flat shipping
fee of $12. Let A(x) = the total cost of an online order of x boxes
of apples, i.e. the cost of the apples plus the $12 shipping fee,
what is A(x)?
The total cost function is the cost of the apples, i.e. 8x, plus the
flat shipping fee of $12, so that A(x) = 8x + 12.
c. What does A(7) mean and what is A(7)?

37. Notation and Algebra of Functions
The above algebra which utilizes the function notation enables
us to keep track of quantitative information efficiently.
Example B.
a. Apples cost $8.00/box, what is the cost of x boxes of apples?
At the $8.00/box, x boxes of apples cost 8x.
b. For an online order of boxes of apples there is a flat shipping
fee of $12. Let A(x) = the total cost of an online order of x boxes
of apples, i.e. the cost of the apples plus the $12 shipping fee,
what is A(x)?
The total cost function is the cost of the apples, i.e. 8x, plus the
flat shipping fee of $12, so that A(x) = 8x + 12.
c. What does A(7) mean and what is A(7)?
A(7) is the “cost of an online order with x = 7 or a 7–box apple
order”

38. Notation and Algebra of Functions
The above algebra which utilizes the function notation enables
us to keep track of quantitative information efficiently.
Example B.
a. Apples cost $8.00/box, what is the cost of x boxes of apples?
At the $8.00/box, x boxes of apples cost 8x.
b. For an online order of boxes of apples there is a flat shipping
fee of $12. Let A(x) = the total cost of an online order of x boxes
of apples, i.e. the cost of the apples plus the $12 shipping fee,
what is A(x)?
The total cost function is the cost of the apples, i.e. 8x, plus the
flat shipping fee of $12, so that A(x) = 8x + 12.
c. What does A(7) mean and what is A(7)?
A(7) is the “cost of an online order with x = 7 or a 7–box apple
order” and A(7) = 8(7) + 12 = 64.

39. Notation and Algebra of Functions
The above algebra which utilizes the function notation enables
us to keep track of quantitative information efficiently.
Example B.
a. Apples cost $8.00/box, what is the cost of x boxes of apples?
At the $8.00/box, x boxes of apples cost 8x.
b. For an online order of boxes of apples there is a flat shipping
fee of $12. Let A(x) = the total cost of an online order of x boxes
of apples, i.e. the cost of the apples plus the $12 shipping fee,
what is A(x)?
The total cost function is the cost of the apples, i.e. 8x, plus the
flat shipping fee of $12, so that A(x) = 8x + 12.
c. What does A(7) mean and what is A(7)?
A(7) is the “cost of an online order with x = 7 or a 7–box apple
order” and A(7) = 8(7) + 12 = 64. Hence the total cost is $64
for an online order of 7 boxes of apples.

40. Notation and Algebra of Functions
d. Banana cost $6.00/box. One online order of y boxes of
bananas has a flat shipping fee of $20, let B(y) = total cost of
an online order of y boxes of bananas find B(y). What does
B(9) mean? and what is B(9)?

41. Notation and Algebra of Functions
d. Banana cost $6.00/box. One online order of y boxes of
bananas has a flat shipping fee of $20, let B(y) = total cost of
an online order of y boxes of bananas find B(y). What does
B(9) mean? and what is B(9)?
At the $6.00/box, y boxes of bananas cost 6y,

42. Notation and Algebra of Functions
d. Banana cost $6.00/box. One online order of y boxes of
bananas has a flat shipping fee of $20, let B(y) = total cost of
an online order of y boxes of bananas find B(y). What does
B(9) mean? and what is B(9)?
At the $6.00/box, y boxes of bananas cost 6y,
plus the flat shipping fee of $20, the total cost function for y
boxes of bananas is B(y) = 6y + 20.

43. Notation and Algebra of Functions
d. Banana cost $6.00/box. One online order of y boxes of
bananas has a flat shipping fee of $20, let B(y) = total cost of
an online order of y boxes of bananas find B(y). What does
B(9) mean? and what is B(9)?
At the $6.00/box, y boxes of bananas cost 6y,
plus the flat shipping fee of $20, the total cost function for y
boxes of bananas is B(y) = 6y + 20.
B(9) is the cost of an online order for 9 boxes of bananas

44. Notation and Algebra of Functions
d. Banana cost $6.00/box. One online order of y boxes of
bananas has a flat shipping fee of $20, let B(y) = total cost of
an online order of y boxes of bananas find B(y). What does
B(9) mean? and what is B(9)?
At the $6.00/box, y boxes of bananas cost 6y,
plus the flat shipping fee of $20, the total cost function for y
boxes of bananas is B(y) = 6y + 20.
B(9) is the cost of an online order for 9 boxes of bananas and
B(9) = 6(9) + 20 = 74

45. Notation and Algebra of Functions
d. Banana cost $6.00/box. One online order of y boxes of
bananas has a flat shipping fee of $20, let B(y) = total cost of
an online order of y boxes of bananas find B(y). What does
B(9) mean? and what is B(9)?
At the $6.00/box, y boxes of bananas cost 6y,
plus the flat shipping fee of $20, the total cost function for y
boxes of bananas is B(y) = 6y + 20.
B(9) is the cost of an online order for 9 boxes of bananas and
B(9) = 6(9) + 20 = 74
e. We make three separate orders of 7 boxes of apples and two
orders of 9 boxes of bananas, how much more does it cost for
the apple-orders than the banana-orders? Write the answer
using the function notation. Evaluate.

46. Notation and Algebra of Functions
d. Banana cost $6.00/box. One online order of y boxes of
bananas has a flat shipping fee of $20, let B(y) = total cost of
an online order of y boxes of bananas find B(y). What does
B(9) mean? and what is B(9)?
At the $6.00/box, y boxes of bananas cost 6y,
plus the flat shipping fee of $20, the total cost function for y
boxes of bananas is B(y) = 6y + 20.
B(9) is the cost of an online order for 9 boxes of bananas and
B(9) = 6(9) + 20 = 74
e. We make three separate orders of 7 boxes of apples and two
orders of 9 boxes of bananas, how much more does it cost for
the apple-orders than the banana-orders? Write the answer
using the function notation. Evaluate.
A 7–box apple order is A(7) and a 9–box banana–order is B(9).

47. Notation and Algebra of Functions
d. Banana cost $6.00/box. One online order of y boxes of
bananas has a flat shipping fee of $20, let B(y) = total cost of
an online order of y boxes of bananas find B(y). What does
B(9) mean? and what is B(9)?
At the $6.00/box, y boxes of bananas cost 6y,
plus the flat shipping fee of $20, the total cost function for y
boxes of bananas is B(y) = 6y + 20.
B(9) is the cost of an online order for 9 boxes of bananas and
B(9) = 6(9) + 20 = 74
e. We make three separate orders of 7 boxes of apples and two
orders of 9 boxes of bananas, how much more does it cost for
the apple-orders than the banana-orders? Write the answer
using the function notation. Evaluate.
A 7–box apple order is A(7) and a 9–box banana–order is B(9).
So the difference between the cost of three apple–orders and
two banana–order is 3*A(7) 2*B(9)

48. Notation and Algebra of Functions
d. Banana cost $6.00/box. One online order of y boxes of
bananas has a flat shipping fee of $20, let B(y) = total cost of
an online order of y boxes of bananas find B(y). What does
B(9) mean? and what is B(9)?
At the $6.00/box, y boxes of bananas cost 6y,
plus the flat shipping fee of $20, the total cost function for y
boxes of bananas is B(y) = 6y + 20.
B(9) is the cost of an online order for 9 boxes of bananas and
B(9) = 6(9) + 20 = 74
e. We make three separate orders of 7 boxes of apples and two
orders of 9 boxes of bananas, how much more does it cost for
the apple-orders than the banana-orders? Write the answer
using the function notation. Evaluate.
A 7–box apple order is A(7) and a 9–box banana–order is B(9).
So the difference between the cost of three apple–orders and
two banana–order is 3*A(7) – 2*B(9)

49. Notation and Algebra of Functions
d. Banana cost $6.00/box. One online order of y boxes of
bananas has a flat shipping fee of $20, let B(y) = total cost of
an online order of y boxes of bananas find B(y). What does
B(9) mean? and what is B(9)?
At the $6.00/box, y boxes of bananas cost 6y,
plus the flat shipping fee of $20, the total cost function for y
boxes of bananas is B(y) = 6y + 20.
B(9) is the cost of an online order for 9 boxes of bananas and
B(9) = 6(9) + 20 = 74
e. We make three separate orders of 7 boxes of apples and two
orders of 9 boxes of bananas, how much more does it cost for
the apple-orders than the banana-orders? Write the answer
using the function notation. Evaluate.
A 7–box apple order is A(7) and a 9–box banana–order is B(9).
So the difference between the cost of three apple–orders and
two banana–order is 3*A(7) – 2*B(9) =

50. Notation and Algebra of Functions
d. Banana cost $6.00/box. One online order of y boxes of
bananas has a flat shipping fee of $20, let B(y) = total cost of
an online order of y boxes of bananas find B(y). What does
B(9) mean? and what is B(9)?
At the $6.00/box, y boxes of bananas cost 6y,
plus the flat shipping fee of $20, the total cost function for y
boxes of bananas is B(y) = 6y + 20.
B(9) is the cost of an online order for 9 boxes of bananas and
B(9) = 6(9) + 20 = 74
e. We make three separate orders of 7 boxes of apples and two
orders of 9 boxes of bananas, how much more does it cost for
the apple-orders than the banana-orders? Write the answer
using the function notation. Evaluate.
A 7–box apple order is A(7) and a 9–box banana–order is B(9).
So the difference between the cost of three apple–orders and
two banana–order is 3*A(7) – 2*B(9) = 3*68 – 2*74 = $56.

51. Notation and Algebra of Functions
The input of a formula–function may be other mathematics
expressions.

52. Notation and Algebra of Functions
The input of a formula–function may be other mathematics
expressions. Often in such problems we are to simplify the
outputs algebraically.

53. Notation and Algebra of Functions
The input of a formula–function may be other mathematics
expressions. Often in such problems we are to simplify the
outputs algebraically.
We write down the Square–Formula as a reminder below.
(The Square–Formula) (a ± b)2 = a2 ± 2ab + b2

54. Notation and Algebra of Functions
The input of a formula–function may be other mathematics
expressions. Often in such problems we are to simplify the
outputs algebraically.
We write down the Square–Formula as a reminder below.
(The Square–Formula) (a ± b)2 = a2 ± 2ab + b2
Example C.
Given f(x) = x2 – 2x + 3, simplify the following.
a. f(2a)

55. Notation and Algebra of Functions
The input of a formula–function may be other mathematics
expressions. Often in such problems we are to simplify the
outputs algebraically.
We write down the Square–Formula as a reminder below.
(The Square–Formula) (a ± b)2 = a2 ± 2ab + b2
Example C.
Given f(x) = x2 – 2x + 3, simplify the following.
a. f(2a) = (2a)2 – 2(2a) + 3

56. Notation and Algebra of Functions
The input of a formula–function may be other mathematics
expressions. Often in such problems we are to simplify the
outputs algebraically.
We write down the Square–Formula as a reminder below.
(The Square–Formula) (a ± b)2 = a2 ± 2ab + b2
Example C.
Given f(x) = x2 – 2x + 3, simplify the following.
a. f(2a) = (2a)2 – 2(2a) + 3
= 4a2 – 4a + 3

57. Notation and Algebra of Functions
The input of a formula–function may be other mathematics
expressions. Often in such problems we are to simplify the
outputs algebraically.
We write down the Square–Formula as a reminder below.
(The Square–Formula) (a ± b)2 = a2 ± 2ab + b2
Example C.
Given f(x) = x2 – 2x + 3, simplify the following.
a. f(2a) = (2a)2 – 2(2a) + 3
= 4a2 – 4a + 3
b. f(a + b) =

58. Notation and Algebra of Functions
The input of a formula–function may be other mathematics
expressions. Often in such problems we are to simplify the
outputs algebraically.
We write down the Square–Formula as a reminder below.
(The Square–Formula) (a ± b)2 = a2 ± 2ab + b2
Example C.
Given f(x) = x2 – 2x + 3, simplify the following.
a. f(2a) = (2a)2 – 2(2a) + 3
= 4a2 – 4a + 3
b. f(a + b) = (a + b)2 – 2(a + b) + 3

59. Notation and Algebra of Functions
The input of a formula–function may be other mathematics
expressions. Often in such problems we are to simplify the
outputs algebraically.
We write down the Square–Formula as a reminder below.
(The Square–Formula) (a ± b)2 = a2 ± 2ab + b2
Example C.
Given f(x) = x2 – 2x + 3, simplify the following.
a. f(2a) = (2a)2 – 2(2a) + 3
= 4a2 – 4a + 3
b. f(a + b) = (a + b)2 – 2(a + b) + 3 by the squaring formula
= a2 + 2ab + b2 – 2a – 2b + 3

60. Notation and Algebra of Functions
The input of a formula–function may be other mathematics
expressions. Often in such problems we are to simplify the
outputs algebraically.
We write down the Square–Formula as a reminder below.
(The Square–Formula) (a ± b)2 = a2 ± 2ab + b2
Example C.
Given f(x) = x2 – 2x + 3, simplify the following.
a. f(2a) = (2a)2 – 2(2a) + 3
= 4a2 – 4a + 3
b. f(a + b) = (a + b)2 – 2(a + b) + 3 by the squaring formula
= a2 + 2ab + b2 – 2a – 2b + 3
c. f(a + b) – f(2a)

61. Notation and Algebra of Functions
The input of a formula–function may be other mathematics
expressions. Often in such problems we are to simplify the
outputs algebraically.
We write down the Square–Formula as a reminder below.
(The Square–Formula) (a ± b)2 = a2 ± 2ab + b2
Example C.
Given f(x) = x2 – 2x + 3, simplify the following.
a. f(2a) = (2a)2 – 2(2a) + 3
= 4a2 – 4a + 3
b. f(a + b) = (a + b)2 – 2(a + b) + 3 by the squaring formula
= a2 + 2ab + b2 – 2a – 2b + 3
c. f(a + b) – f(2a) from part b and part a
= a2 + 2ab + b2 – 2a – 2b + 3 – (4a2 – 4a + 3)

62. Notation and Algebra of Functions
The input of a formula–function may be other mathematics
expressions. Often in such problems we are to simplify the
outputs algebraically.
We write down the Square–Formula as a reminder below.
(The Square–Formula) (a ± b)2 = a2 ± 2ab + b2
Example C.
Given f(x) = x2 – 2x + 3, simplify the following.
a. f(2a) = (2a)2 – 2(2a) + 3
= 4a2 – 4a + 3
b. f(a + b) = (a + b)2 – 2(a + b) + 3 by the squaring formula
= a2 + 2ab + b2 – 2a – 2b + 3
c. f(a + b) – f(2a) from part b and part a
= a2 + 2ab + b2 – 2a – 2b + 3 – (4a2 – 4a + 3)
= –3a2 + 2ab + b2 – 2b – 6a + 6

63. Notation and Algebra of Functions
d. Simplify f(x + h) – f(x)

64. Notation and Algebra of Functions
d. Simplify f(x + h) – f(x)
Given that f(x) = x2 – 2x + 3, we calculate that
f(x + h ) =

65. Notation and Algebra of Functions
d. Simplify f(x + h) – f(x)
Given that f(x) = x2 – 2x + 3, we calculate that
f(x + h ) = (x+h)2 – 2(x+h) – 3

66. Notation and Algebra of Functions
d. Simplify f(x + h) – f(x)
Given that f(x) = x2 – 2x + 3, we calculate that
f(x + h ) = (x+h)2 – 2(x+h) – 3
= x2 + 2xh + h2 – 2x – 2h – 3

67. Notation and Algebra of Functions
d. Simplify f(x + h) – f(x)
Given that f(x) = x2 – 2x + 3, we calculate that
f(x + h ) = (x+h)2 – 2(x+h) – 3
= x2 + 2xh + h2 – 2x – 2h – 3
Hence f(x+h) – f(x)

68. Notation and Algebra of Functions
d. Simplify f(x + h) – f(x)
Given that f(x) = x2 – 2x + 3, we calculate that
f(x + h ) = (x+h)2 – 2(x+h) – 3
= x2 + 2xh + h2 – 2x – 2h – 3
Hence f(x+h) – f(x)
= x2 + 2xh + h2 – 2x – 2h – 3 –

69. Notation and Algebra of Functions
d. Simplify f(x + h) – f(x)
Given that f(x) = x2 – 2x + 3, we calculate that
f(x + h ) = (x+h)2 – 2(x+h) – 3
= x2 + 2xh + h2 – 2x – 2h – 3
Hence f(x+h) – f(x)
= x2 + 2xh + h2 – 2x – 2h – 3 – (x2 – 2x – 3)

70. Notation and Algebra of Functions
d. Simplify f(x + h) – f(x)
Given that f(x) = x2 – 2x + 3, we calculate that
f(x + h ) = (x+h)2 – 2(x+h) – 3
= x2 + 2xh + h2 – 2x – 2h – 3
Hence f(x+h) – f(x)
= x2 + 2xh + h2 – 2x – 2h – 3 – (x2 – 2x – 3)
= x2 + 2xh + h2 – 2x – 2h – 3 – x2 + 2x + 3

71. Notation and Algebra of Functions
d. Simplify f(x + h) – f(x)
Given that f(x) = x2 – 2x + 3, we calculate that
f(x + h ) = (x+h)2 – 2(x+h) – 3
= x2 + 2xh + h2 – 2x – 2h – 3
Hence f(x+h) – f(x)
= x2 + 2xh + h2 – 2x – 2h – 3 – (x2 – 2x – 3)
= x2 + 2xh + h2 – 2x – 2h – 3 – x2 + 2x + 3
= 2xh + h2 – 2h

72. Notation and Algebra of Functions
d. Simplify f(x + h) – f(x)
Given that f(x) = x2 – 2x + 3, we calculate that
f(x + h ) = (x+h)2 – 2(x+h) – 3
= x2 + 2xh + h2 – 2x – 2h – 3
Hence f(x+h) – f(x)
= x2 + 2xh + h2 – 2x – 2h – 3 – (x2 – 2x – 3)
= x2 + 2xh + h2 – 2x – 2h – 3 – x2 + 2x + 3
= 2xh + h2 – 2h
We may form new functions by using algebraic operations.

73. Notation and Algebra of Functions
d. Simplify f(x + h) – f(x)
Given that f(x) = x2 – 2x + 3, we calculate that
f(x + h ) = (x+h)2 – 2(x+h) – 3
= x2 + 2xh + h2 – 2x – 2h – 3
Hence f(x+h) – f(x)
= x2 + 2xh + h2 – 2x – 2h – 3 – (x2 – 2x – 3)
= x2 + 2xh + h2 – 2x – 2h – 3 – x2 + 2x + 3
= 2xh + h2 – 2h
We may form new functions by using algebraic operations.
Example D. Let f(x) = 3x – 4, g(x) = x2 – 2x – 3,
simplify the following expressions.
(f + g)(x) =
(f – g)(x) =
(f * g)(x) =
(f / g)(x) =
f2(x) =

74. Notation and Algebra of Functions
d. Simplify f(x + h) – f(x)
Given that f(x) = x2 – 2x + 3, we calculate that
f(x + h ) = (x+h)2 – 2(x+h) – 3
= x2 + 2xh + h2 – 2x – 2h – 3
Hence f(x+h) – f(x)
= x2 + 2xh + h2 – 2x – 2h – 3 – (x2 – 2x – 3)
= x2 + 2xh + h2 – 2x – 2h – 3 – x2 + 2x + 3
= 2xh + h2 – 2h
We may form new functions by using algebraic operations.
Example D. Let f(x) = 3x – 4, g(x) = x2 – 2x – 3,
simplify the following expressions.
(f + g)(x) = 3x – 4 + x2 – 2x – 3
(f – g)(x) =
(f * g)(x) =
(f / g)(x) =
f2(x) =

75. Notation and Algebra of Functions
d. Simplify f(x + h) – f(x)
Given that f(x) = x2 – 2x + 3, we calculate that
f(x + h ) = (x+h)2 – 2(x+h) – 3
= x2 + 2xh + h2 – 2x – 2h – 3
Hence f(x+h) – f(x)
= x2 + 2xh + h2 – 2x – 2h – 3 – (x2 – 2x – 3)
= x2 + 2xh + h2 – 2x – 2h – 3 – x2 + 2x + 3
= 2xh + h2 – 2h
We may form new functions by using algebraic operations.
Example D. Let f(x) = 3x – 4, g(x) = x2 – 2x – 3,
simplify the following expressions.
(f + g)(x) = 3x – 4 + x2 – 2x – 3 = x2 + x – 7
(f – g)(x) =
(f * g)(x) =
(f / g)(x) =
f2(x) =

76. Notation and Algebra of Functions
d. Simplify f(x + h) – f(x)
Given that f(x) = x2 – 2x + 3, we calculate that
f(x + h ) = (x+h)2 – 2(x+h) – 3
= x2 + 2xh + h2 – 2x – 2h – 3
Hence f(x+h) – f(x)
= x2 + 2xh + h2 – 2x – 2h – 3 – (x2 – 2x – 3)
= x2 + 2xh + h2 – 2x – 2h – 3 – x2 + 2x + 3
= 2xh + h2 – 2h
We may form new functions by using algebraic operations.
Example D. Let f(x) = 3x – 4, g(x) = x2 – 2x – 3,
simplify the following expressions.
(f + g)(x) = 3x – 4 + x2 – 2x – 3 = x2 + x – 7
(f – g)(x) = 3x – 4 – (x2 – 2x – 3)
(f * g)(x) =
(f / g)(x) =
f2(x) =

77. Notation and Algebra of Functions
d. Simplify f(x + h) – f(x)
Given that f(x) = x2 – 2x + 3, we calculate that
f(x + h ) = (x+h)2 – 2(x+h) – 3
= x2 + 2xh + h2 – 2x – 2h – 3
Hence f(x+h) – f(x)
= x2 + 2xh + h2 – 2x – 2h – 3 – (x2 – 2x – 3)
= x2 + 2xh + h2 – 2x – 2h – 3 – x2 + 2x + 3
= 2xh + h2 – 2h
We may form new functions by using algebraic operations.
Example D. Let f(x) = 3x – 4, g(x) = x2 – 2x – 3,
simplify the following expressions.
(f + g)(x) = 3x – 4 + x2 – 2x – 3 = x2 + x – 7
(f – g)(x) = 3x – 4 – (x2 – 2x – 3) = –x2 + 5x – 1
(f * g)(x) =
(f / g)(x) =
f2(x) =

78. Notation and Algebra of Functions
d. Simplify f(x + h) – f(x)
Given that f(x) = x2 – 2x + 3, we calculate that
f(x + h ) = (x+h)2 – 2(x+h) – 3
= x2 + 2xh + h2 – 2x – 2h – 3
Hence f(x+h) – f(x)
= x2 + 2xh + h2 – 2x – 2h – 3 – (x2 – 2x – 3)
= x2 + 2xh + h2 – 2x – 2h – 3 – x2 + 2x + 3
= 2xh + h2 – 2h
We may form new functions by using algebraic operations.
Example D. Let f(x) = 3x – 4, g(x) = x2 – 2x – 3,
simplify the following expressions.
(f + g)(x) = 3x – 4 + x2 – 2x – 3 = x2 + x – 7
(f – g)(x) = 3x – 4 – (x2 – 2x – 3) = –x2 + 5x – 1
(f * g)(x) = (3x – 4) (x2 – 2x – 3) = 3x3 – 10x2 – x + 12
*
(f / g)(x) =
f2(x) =

79. Notation and Algebra of Functions
d. Simplify f(x + h) – f(x)
Given that f(x) = x2 – 2x + 3, we calculate that
f(x + h ) = (x+h)2 – 2(x+h) – 3
= x2 + 2xh + h2 – 2x – 2h – 3
Hence f(x+h) – f(x)
= x2 + 2xh + h2 – 2x – 2h – 3 – (x2 – 2x – 3)
= x2 + 2xh + h2 – 2x – 2h – 3 – x2 + 2x + 3
= 2xh + h2 – 2h
We may form new functions by using algebraic operations.
Example D. Let f(x) = 3x – 4, g(x) = x2 – 2x – 3,
simplify the following expressions.
(f + g)(x) = 3x – 4 + x2 – 2x – 3 = x2 + x – 7
(f – g)(x) = 3x – 4 – (x2 – 2x – 3) = –x2 + 5x – 1
(f * g)(x) = (3x – 4) (x2 – 2x – 3) = 3x3 – 10x2 – x + 12
*
(f / g)(x) = (3x – 4)/(x2 – 2x – 3)
f2(x) =

80. Notation and Algebra of Functions
d. Simplify f(x + h) – f(x)
Given that f(x) = x2 – 2x + 3, we calculate that
f(x + h ) = (x+h)2 – 2(x+h) – 3
= x2 + 2xh + h2 – 2x – 2h – 3
Hence f(x+h) – f(x)
= x2 + 2xh + h2 – 2x – 2h – 3 – (x2 – 2x – 3)
= x2 + 2xh + h2 – 2x – 2h – 3 – x2 + 2x + 3
= 2xh + h2 – 2h
We may form new functions by using algebraic operations.
Example D. Let f(x) = 3x – 4, g(x) = x2 – 2x – 3,
simplify the following expressions.
(f + g)(x) = 3x – 4 + x2 – 2x – 3 = x2 + x – 7
(f – g)(x) = 3x – 4 – (x2 – 2x – 3) = –x2 + 5x – 1
(f * g)(x) = (3x – 4) (x2 – 2x – 3) = 3x3 – 10x2 – x + 12
*
(f / g)(x) = (3x – 4)/(x2 – 2x – 3)
f2(x) = (3x – 4)2 = 9x2 – 24x + 16

81. Notation and Algebra of Functions
From example B, we have that
A(x) = 8x + 12 – the cost of an x–box order of apples and
B(y) = 9y + 20– the cost of a y–box order of bananas.

82. Notation and Algebra of Functions
From example B, we have that
A(x) = 8x + 12 – the cost of an x–box order of apples and
B(y) = 9y + 20– the cost of a y–box order of bananas.
Note the necessity of two variables x and y because the
number of boxes of apples may be different from the number
of boxes of bananas that we ordered.

83. Notation and Algebra of Functions
From example B, we have that
A(x) = 8x + 12 – the cost of an x–box order of apples and
B(y) = 9y + 20– the cost of a y–box order of bananas.
Note the necessity of two variables x and y because the
number of boxes of apples may be different from the number
of boxes of bananas that we ordered.
However, if we are only to order the same number of boxes of
apples as bananas, then a single variable x is suffice,

84. Notation and Algebra of Functions
From example B, we have that
A(x) = 8x + 12 – the cost of an x–box order of apples and
B(y) = 9y + 20– the cost of a y–box order of bananas.
Note the necessity of two variables x and y because the
number of boxes of apples may be different from the number
of boxes of bananas that we ordered.
However, if we are only to order the same number of boxes of
apples as bananas, then a single variable x is suffice, i.e.
A(x) = cost of an x–box order of apples and
B(x) = 6x + 20–cost of an x–box order of bananas,

85. Notation and Algebra of Functions
From example B, we have that
A(x) = 8x + 12 – the cost of an x–box order of apples and
B(y) = 9y + 20– the cost of a y–box order of bananas.
Note the necessity of two variables x and y because the
number of boxes of apples may be different from the number
of boxes of bananas that we ordered.
However, if we are only to order the same number of boxes of
apples as bananas, then a single variable x is suffice, i.e.
A(x) = cost of an x–box order of apples and
B(x) = 6x + 20–cost of an x–box order of bananas,
Example E. Given that x = the number of boxes in one order,
A(x) = 8x + 12 and that B(x) = 9x + 20. Simplify the following
expressions and what do they signify.
a. (A + B)(x)

86. Notation and Algebra of Functions
From example B, we have that
A(x) = 8x + 12 – the cost of an x–box order of apples and
B(y) = 9y + 20– the cost of a y–box order of bananas.
Note the necessity of two variables x and y because the
number of boxes of apples may be different from the number
of boxes of bananas that we ordered.
However, if we are only to order the same number of boxes of
apples as bananas, then a single variable x is suffice, i.e.
A(x) = cost of an x–box order of apples and
B(x) = 6x + 20–cost of an x–box order of bananas,
Example E. Given that x = the number of boxes in one order,
A(x) = 8x + 12 and that B(x) = 9x + 20. Simplify the following
expressions and what do they signify.
a. (A + B)(x)
(A + B)(x) = (8x + 12) + (6x + 20) = 14x + 32

87. Notation and Algebra of Functions
From example B, we have that
A(x) = 8x + 12 – the cost of an x–box order of apples and
B(y) = 9y + 20– the cost of a y–box order of bananas.
Note the necessity of two variables x and y because the
number of boxes of apples may be different from the number
of boxes of bananas that we ordered.
However, if we are only to order the same number of boxes of
apples as bananas, then a single variable x is suffice, i.e.
A(x) = cost of an x–box order of apples and
B(x) = 6x + 20–cost of an x–box order of bananas,
Example E. Given that x = the number of boxes in one order,
A(x) = 8x + 12 and that B(x) = 9x + 20. Simplify the following
expressions and what do they signify.
a. (A + B)(x)
(A + B)(x) = (8x + 12) + (6x + 20) = 14x + 32 is the total cost of
one order of x–boxes of apples and x–boxes of bananas.

88. Notation and Algebra of Functions
b. Simplify (3A – 2B)(x). What is (3A – 2B)(5) and what does it
mean?

89. Notation and Algebra of Functions
b. Simplify (3A – 2B)(x). What is (3A – 2B)(5) and what does it
mean?
(3A – 2B)(x) =

90. Notation and Algebra of Functions
b. Simplify (3A – 2B)(x). What is (3A – 2B)(5) and what does it
mean?
(3A – 2B)(x) = 3(8x + 12) – 2(6x + 20)

91. Notation and Algebra of Functions
b. Simplify (3A – 2B)(x). What is (3A – 2B)(5) and what does it
mean?
(3A – 2B)(x) = 3(8x + 12) – 2(6x + 20) = 12x – 4

92. Notation and Algebra of Functions
b. Simplify (3A – 2B)(x). What is (3A – 2B)(5) and what does it
mean?
(3A – 2B)(x) = 3(8x + 12) – 2(6x + 20) = 12x – 4 is the
difference in cost of three orders of x–boxes of apples
versus two orders of x–boxes of bananas.

93. Notation and Algebra of Functions
b. Simplify (3A – 2B)(x). What is (3A – 2B)(5) and what does it
mean?
(3A – 2B)(x) = 3(8x + 12) – 2(6x + 20) = 12x – 4 is the
difference in cost of three orders of x–boxes of apples
versus two orders of x–boxes of bananas.
(3A – 2B)(5) = 12(5) – 4 =

94. Notation and Algebra of Functions
b. Simplify (3A – 2B)(x). What is (3A – 2B)(5) and what does it
mean?
(3A – 2B)(x) = 3(8x + 12) – 2(6x + 20) = 12x – 4 is the
difference in cost of three orders of x–boxes of apples
versus two orders of x–boxes of bananas.
(3A – 2B)(5) = 12(5) – 4 = $56

95. Notation and Algebra of Functions
b. Simplify (3A – 2B)(x). What is (3A – 2B)(5) and what does it
mean?
(3A – 2B)(x) = 3(8x + 12) – 2(6x + 20) = 12x – 4 is the
difference in cost of three orders of x–boxes of apples
versus two orders of x–boxes of bananas.
(3A – 2B)(5) = 12(5) – 4 = $56 is the difference in cost of three
orders of 5–boxes of apples versus two orders of 5–boxes of
bananas.