1.6E: Inverse Functions
( \newcommand{\kernel}{\mathrm{null}\,}\)
Assume that the function f is a one-to-one function.
1. If f(6)=7, find f^{-1} (7)
2. If f(3)=2, find f^{-1} (2)
3. If f^{-1} (-4)=-8, find f(-8)
4. If f^{-1} (-2)=-1, find f(-1)
5. If f(5)=2, find (f(5))^{-1}
6. If f(1)=4, find (f(1))^{-1}
7. Using the graph of f(x) shown
a. Find f(0)
b. Solve f(x)=0
c. Find f^{-1} (0)
d. Solve f^{-1} (x)=0
8. Using the graph shown
a. Find g(1)
b. Solve g(x)=1
c. Find g^{-1} (1)
d. Solve g^{-1} (x)=1
9. Use the table below to find the indicated quantities.
x | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
f(x) | 8 | 0 | 7 | 4 | 2 | 6 | 5 | 3 | 9 | 1 |
a. Find f(1)
b. Solve f(x)=3
c. Find f^{-1}(0)
d. Solve f^{-1}(x)=7
10. Use the table below to fill in the missing values.
t | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
h(t) | 6 | 0 | 1 | 7 | 2 | 3 | 5 | 4 | 9 |
a. Find h(6)
b. Solve h(t)=0
c. Find h^{-1} (5)
d. Solve h^{-1} (t)=1
For each table below, create a table for f^{-1} (x).
11.
x | 3 | 6 | 9 | 13 | 14 |
f(x) | 1 | 4 | 7 | 12 | 16 |
For each function below, find f^{-1} (x)
13. f(x)=x+3
14. f(x)=x+5
15. f(x)= 2 - x
16. f(x)=3-x
17. f(x)=11x+7
18. f(x)=9+10x
For each function, find a domain on which f is one-to-one and non-decreasing, then find the inverse of f restricted to that domain.
19. f(x)=(x +7)^{2}
20. f(x)=(x-6)^{2}
21. f(x)=x^{2} -5
22. f(x)=x^{2} +1
23. If f(x)=x^{3} -5 and g(x)=\sqrt[{3}]{x+5}, find
a. f(g(x))
b. g(f(x))
c. What does this tell us about the relationship between f(x) and g(x)?
24. If f(x)=\dfrac{x}{2+x} and g(x)=\dfrac{2x}{1-x}, find
a. f(g(x))
b. g(f(x))
c. What does this tell us about the relationship between f(x) and g(x)?
- Answer
-
1. 6
3. -4
5. 1/2
7a. 3
b. 2
c. 2
d. 211.
x 1 4 7 12 16 f^{-1}(x) 3 6 9 13 14 13. f^{-1}(x) = x -3
15. f^{-1}(x) = -x + 2
17. f^{-1}(x) = \dfrac{x - 7}{11}
19. Restricted domain x \ge -7, f^{-1}(x) = \sqrt{x} - 7
21. Restricted domain x \ge 0, f^{-1}(x) = \sqrt{x + 5}
23a. f(g(x)) = (\sqrt[3]{x + 5})^3 - 5 = x
b. g(f(x)) = \sqrt[3]{x^3 - 5 + 5} = x
c. This means that they are inverse functions (of each other)