The document discusses operations on functions including: - Finding the domain of various rational functions - Graphing a rational function in quadrant II, stating its domain and range, and determining asymptotes - Defining even and odd functions graphically and symbolically - Examples of determining whether functions are even or odd based on their behavior under substitution of -x

1. Operations on Functions
Surgeons

2. and their domains.
For each of the following find
Domain:
Domain:
Domain:
Domain:

3. and their domains.
For each of the following find
Domain:
Domain:
Domain:
Domain:

4. and their domains.
For each of the following find
Domain:
Domain:
Domain:
Domain:

5. Using your calculator, find the coordinates of the the quot;turning pointquot; of ƒ in
quadrant II.

6. Graph the rational function ƒ. State the domain and range and determine left-
and right-hand asymptotes numerically.

7. Graph the rational function ƒ. State the domain and range and determine left-
and right-hand asymptotes numerically.

8. a little review ...

9. EVEN FUNCTIONS
Graphically: A function is quot;evenquot; if its graph is symmetrical about the y-axis.
These functions
are even...
These are
not ...
Symbolically (Algebraically)
a function is quot;evenquot; IFF (if and only if) ƒ(-x) = ƒ(x)
Examples: Are these functions even?
1. f(x) = x² 2. g(x) = x² + 2x
f(-x) = (-x)² g(-x) = (-x)² + 2(-x)
f(-x) = x² g(-x) = x² - 2x
since f(-x)=f(x) since g(-x) is not equal to g(x)
f is an even function g is not an even function

10. ODD FUNCTIONS
Graphically: A function is quot;oddquot; if its graph is symmetrical about the origin.
These
functions
These are
are odd ... not ...
Symbolically (Algebraically)
a function is quot;oddquot; IFF (if and only if) ƒ(-x) = -ƒ(x)
Examples: 1. ƒ(x) = x³ - x 2. g(x) = x³- x²
ƒ(-x) = (-x)³ - (-x) g(-x) = (-x)³ - (-x)²
ƒ(x) = -x³ + x g(x) = -x³ - x²
-ƒ(x) = -(x³ - x) -g(x) = -(x³-x²)
-ƒ(x) = -x³ + x -g(x) = -x³+ x²
since ƒ(-x)= -ƒ(x) since g(-x) is not equal to -g(x)
ƒ is an odd function g is not an odd function

11. Demonstrate, in as many different ways as you can think of, whether the
function ƒ is even, odd or neither.