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# 1.6.6E: Inverse Functions

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section 1.6 exercises

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 $$g(x)$$?

1. 6

3. -4

5. 1/2

7a. 3
b. 2
c. 2
d. 2

11.

 $$x$$ 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)