3. INVERSE FUNCTIONS
• In short, the reflector of the original
function at the radical axis y = x
• The original function is
f(x)
and then the inverse function of f(x) is:
f-1(x) or F(x) in other books
4. INVERSE FUNCTIONS
• In terms of ordered pairs, the inverse of
f(x) = (a,b) is
f-1(x) = F(x) (b,a)
• In short, the inverse of the set:
f(x) = (a1,b1), (a2,b2), (a3,b3),…, (an+1,bn+1)
is
f-1 (x) = F(x) = (b1, a1), (b2,a2), (b3,a3),…,
(bn+1,an+1)
6. y = f(x)
(a1, b1)
(a2, b2)
(a3, b3)
(an+1, bn+1)
(b1, a1)
(b2, a2)
(b3, a3)
(bn+1, an+1)
The inverse of
f(x):
f-1(x) = F(x)
The set of
ordered pairs at
f(x) has been
inverted
7. INVERSE FUNCTIONS
EXAMPLE : FIND THE INVERSE
FUNCTION OF THE FOLLOWING:
f(x) = (-2,-6), (2,-4), (6,-2), (10,0)
Ans:
f-1(x) = (-6,-2), (-4,2), (-2,6), (0,10)
9. INVERSE FUNCTIONS
Now, in terms of POLYNOMIAL FUNCTION. Here are
the steps to get the inverse function [f-1(x)] of the
original function f(x):
1. Change f(x) to “y” on the given function.
2. Invert the variables between x and y. The y variable
in (1) will be “x” and for x variable on right side will
be “y”.
3. Solve for y from (2).
4. Change “y” into f-1(x).
5. Solve for f [f-1(x)] and f-1[f(x)] (Composition Method).
The answer must be “x”.
21. -5 -4 -3 -2 -1 1 2 3 4 5 6
-3
-2
-1
1
2
3
4
5
y
INVERT THE ORDERED
PAIRS FROM f(x) to
graph
(no need to solve)
P1 P2
x 0 3
y 0
P1 P2
x 0
y 0 3 (3,0)
(0, -3/2)
(0,3)
(-3/2, 0)
30. INVERSE FUNCTIONS
POSSIBLE TO GRAPH ?
You may use the graphical software
for Cartesian and Polar coordinates
CLICK HERE
31. INVERSE FUNCTIONS
QUESTIONS?
For graphing: you will graph only linear functions ( y = mx + b).
Other functions like:
exponential (y = bx)
logarithmic (y = logb x or y = ln x) ,
trigonometric (y = a sin x)
and second degree or higher polynomials
(y = axn + xn-1 +…+a0)
are not yet discussed for way of sketching the function, sometimes
you need to use programmable and graphical calculators or the
computers. It’s hard to sketch the mentioned functions.
32.
33. INVERSE FUNCTIONS
If you want this application
program for graphing purposes
install on your Personal
Computer,
visit www.padowan.dk
this is a free-download software
program.
