This document provides examples and exercises on inverse functions. It first shows an example of finding the inverse of the function f(x) = 2x-3/(x+2). It gives the steps to solve for x in terms of y and obtain the inverse function f^-1(x) = -2x-3/(x-2). It then asks the reader to verify that f(f^-1(x)) = x. The exercises provide 20 functions and ask the reader to find their inverses and verify the inverses are correct. It also gives graphs of 8 functions and asks the reader to determine properties of the inverse graphs, including their domains and ranges and any fixed or end points.

1. Example D.
Hence f -1(y) =
2x – 3
x + 2
Inverse Functions
a. Given f(x) = find f -1(x).,
Set y = and solve for x in term of y.
2x – 3
x + 2 ,
Clear the denominator, we get
y(x + 2) = 2x – 3
yx + 2y = 2x – 3 collecting and
isolating xyx – 2x = –2y – 3
(y – 2)x = –2y – 3
x =
–2y – 3
y – 2
–2y – 3
y – 2
Write the answer using x as the variable:
f -1(x) =
–2x – 3
x – 2

2. Inverse Functions
b. Verify that f(f -1(x)) = x
We've f(x) = and
2x – 3
x + 2 , f -1(x) =
–2x – 3
x – 2
f(f -1(x)) = f( )–2x – 3
x – 2
=
–2x – 3
x – 2
– 3
–2x – 3
x – 2
+ 2
( )2[
[ ]
](x – 2)
(x – 2)
=
2(-2x – 3) – 3(x – 2)
(-2x – 3) + 2(x – 2)
=
-4x – 6 – 3x + 6
-2x – 3 + 2x – 4
=
-7x
-7
= x
Your turn. Verify that f -1(f(x)) = x
Use the LCD to simplify
the complex fraction

3. Exercise A. Find the inverse f -1(x) of the following f(x)’s.
Inverse Functions
2. f(x) = x/3 – 2
2x – 3
x + 2
1. f(x) = 3x – 2 3. f(x) = x/3 + 2/5
5. f(x) = 3/x – 24. f(x) = ax + b 6. f(x) = 4/x + 2/5
7. f(x) = 3/(x – 2) 8. f(x) = 4/(x + 2) – 5
9. f(x) = 2/(3x + 4) – 5 10. f(x) = 5/(4x + 3) – 1/2
f(x) =11. 2x – 3
x + 2f(x) =12.
4x – 3
–3x + 2
f(x) =13.
bx + c
a
f(x) =14. cx + d
ax + b
f(x) =15.
16. f(x) = (3x – 2)1/3
18. f(x) = (x/3 – 2)1/3
17. f(x) = (x/3 + 2/5)1/3
19. f(x) = (x/3)1/3 – 2
20. f(x) = (ax – b)1/3 21. f(x) = (ax)1/3 – b
B. Verify your answers are correct by verifying that
f -1(f(x)) = x and f(f-1 (x)) = x for problem 1 – 21.

4. C. For each of the following graphs of f(x)’s, determine
a. the domain and the range of the f -1(x),
b. the end points and the fixed points of the graph of f -1(x).
Draw the graph of f -1(x).
Inverse Functions
(–3, –1)
y = x
(3,4)
1.
(–4, –2)
(5,7)
2.
(–2, –5)
(4,3)
3.
(–3, –1)
(3,4)4.
(–4, –2)
(5,7)
5.
(–2, –5)
(4,3)
6.

5. Inverse Functions
(2, –1)
7.
(–4, 3)
(c, d)
8.
(a, b)
C. For each of the following graphs of f(x)’s, determine
a. the domain and the range of the f -1(x),
b. the end points and the fixed points of the graph of f -1(x).
Draw the graph of f -1(x).

6. (Answers to the odd problems) Exercise A.
Inverse Functions
2x + 3
2 – x
1. f -1(x) = (x + 2) 3. f -1(x) = (5x – 2)
5. f -1(x) = 7. f -1(x) =
9. f -1(x) = – f -1(x) =11.
2x + 3
3x + 4
f -1 (x) =13. x – ac
ad – b
f -1 (x) =15.
17. f -1(x) = (5x3 – 2) 19. f -1(x) = 3 (x3 + 6x2 + 12x + 8)
21. f -1 (x) = (x + a)3/b
1
3
3
5
3
x + 2
2x + 3
x
2(2x + 9)
3(x + 5)
3
5

7. Exercise C.
Inverse Functions
(4, 3)
y = x
(-1, -3)
1. domain: [-1, 4]
range: [-3, 3]
(–5, –2)
(3,4)
3. domain: [-5, 3]
range: [-2, 4]
(–2, –4)
(7,5)
5. domain: [-2, 7]
range: [-4, 5]
(3, –4)
7. domain: [-1, 3], range: [-4,2]
(–1, 2)