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# 1.4.4E: Composition of Functions

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SECTION 1.4 EXERCISE

Given each pair of functions, calculate $$f(g(0))$$ and $$g(f(0))$$.

1. $$f(x) = 4x + 8, g(x) = 7 - x^{2}$$

2. $$f(x) = 5x + 7, g(x) = 4 - 2x^{2}$$

3. $$f(x) = \sqrt{x + 4} , g(x) = 12 - x^{3}$$

4. $$f(x) = \dfrac{1}{x + 2} , g(x) = 4x + 3$$

Use the table of values to evaluate each expression 5. $$f(g(8))$$

6. $$f(g(5))$$

7. $$g(f(5))$$

8. $$g(f(3))$$

9. $$f(f(4))$$

10. $$f(f(1))$$

11. $$g(g(2))$$

12. $$g(g(6))$$

Use the graphs to evaluate the expressions below. 13. $$f(g(3))$$

14. $$f(g(1))$$

15. $$g(f(1))$$

16. $$g(f(0))$$

17. $$f(f(5))$$

18. $$f(f(4))$$

19. $$g(g(2))$$

20. $$g(g(0))$$

For each pair of functions, find $$f(g(x))$$ and $$g(f(x))$$. Simplify your answers.

21. $$f(x) = \dfrac{1}{x - 6}, g(x) = \dfrac{7}{x} + 6$$

22. $$f(x) = \dfrac{1}{x-4} , g(x) = \dfrac{2}{x} + 4$$

23. $$f(x) = x^{2} + 1, g(x) = \sqrt{x+2}$$

24. $$f(x) = \sqrt{x} +2, g(x) = x^{2} +3$$

25. $$f(x) = |x|, g(x) = 5x + 1$$

26. $$f(x)=\sqrt[{3}]{x} , g(x) = \dfrac{x+1}{x^{3} }$$

27. If $$f(x) = x^{4} +6$$, $$g(x) = x - 6$$ and $$h(x) = \sqrt{x}$$, find $$f(g(h(x)))$$

28. If $$f(x) = x^{2} +1$$, $$g(x) = \dfrac{1}{x}$$ and $$h(x) = x + 3$$ , find $$f(g(h(x)))$$

29. The function $$D(p)$$ gives the number of items that will be demanded when the price is $$p$$. The production cost, $$C(x)$$ is the cost of producing $$x$$ items. To determine the cost of production when the price is \$6, you would do which of the following:

a. Evaluate $$D(C(6))$$
b. Evaluate $$C(D(6))$$
c. Solve $$D(C(x)) = 6$$
d. Solve $$C(D(p)) = 6$$

20. The function $$A(d)$$ gives the pain level on a scale of 0-10 experienced by a patient with $$d$$ milligrams of a pain reduction drug in their system. The milligrams of drug in the patient’s system after t minutes is modeled by $$m(t)$$. To determine when the patient will be at a pain level of 4, you would need to:

a. Evaluate $$A(m(4))$$
b. Evaluate $$m(A(4))$$
c. Solve $$A(m(t) = 4$$
d. Solve $$m(A(d)) = 4$$

31. The radius $$r$$, in inches, of a spherical balloon is related to the volume, $$V$$, by $$r(V)=\sqrt[{3}]{\dfrac{3V}{4\pi } }$$. Air is pumped into the balloon, so the volume after $$t$$ seconds is given by $$V(t) = 10 + 20t$$.

a. Find the composite function $$r(V(t))$$

b. Find the radius after 20 seconds

32. The number of bacteria in a refrigerated food product is given by $$N(T) = 23T^{2} - 56T + 1$$, $$3 < T < 33$$, where $$T$$ is the temperature of the food. When the food is removed from the refrigerator, the temperature is given by $$T(t) = 5t + 1.5$$, where t is the time in hours.

a. Find the composite function $$N\left(T\left(t\right)\right)$$
b. Find the bacteria count after 4 hours

33. Given $$p(x) = \dfrac{1}{\sqrt{x} }$$ and $$m(x) = x^{2} -4$$, find the domain of $$m(p(x)$$.

34. Given $$p(x) = \dfrac{1}{\sqrt{x} }$$ and $$m(x) = 9 - x^{2}$$, find the domain of $$m(p(x))$$.

35. Given $$f(x) = \dfrac{1}{x+3}$$ and $$g(x) = \dfrac{2}{x - 1}$$, find the domain of $$f(g(x))$$.

36. Given $$f(x) = \dfrac{x}{x+1}$$ and $$g(x)=\dfrac{4}{x}$$, find the domain of $$f(g(x))$$.

37. Given $$f(x)=\sqrt{x-2}$$ and $$g(x)=\dfrac{2}{x^{2} -3}$$, find the domain of $$g(f(x))$$.

38. Given $$f(x)=\sqrt{4-x}$$ and $$g(x)=\dfrac{1}{x^{2} -2}$$, find the domain of $$g(f(x))$$.

Find functions $$f(x)$$ and $$g(x)$$ so the given function can be expressed as $$h(x)=f(g(x))$$.

39. $$h(x)=(x+2)^{2}$$

40. $$(x)=(x-5)^{3}$$

41. $$(x)=\dfrac{3}{x-5}$$

42. $$h(x)=\dfrac{4}{(x+2)^{2}}$$

43. $$h(x)=3+\sqrt{x-2}$$

44. $$h(x)=4+\sqrt[{3}]{x}$$

45. Let $$f(x)$$ be a linear function, with form $$f(x)=ax+b$$ for constants $$a$$ and $$b$$. [UW]

a. Show that $$f\left(f\left(x\right)\right)$$ is a linear function
b. Find a function $$g(x)$$ such that $$g\left(g\left(x\right)\right)=6x-8$$

46. Let $$f(x)=\dfrac{1}{2} x+3$$ [UW]

a. Sketch the graphs of $$f(x)$$, $$f(f(x))$$, $$f(f(f(x)))$$ on the interval $$-2 \le x \le 10$$
b. Your graphs should all intersect at the point (6, 6). The value x = 6 is called a fixed point of the function $$f(x)$$ since $$f(6) = 6$$; that is, 6 is fixed - it doesn’t move when $$f$$ is applied to it. Give an explanation for why 6 is a fixed point for any function $$f(f(f(...f(x)...)))$$.
c. Linear functions (with the exception of $$f(x)=x$$) can have at most one fixed point. Quadratic functions can have at most two. Find the fixed points of the function $$g(x)=x^{2} -2$$.
d. Give a quadratic function whose fixed points are $$x = -2$$ and $$x = 3$$.

47. A car leaves Seattle heading east. The speed of the car in mph after $$m$$ minutes is given by the function $$C(m)=\dfrac{70m^{2} }{10+m^{2} }$$. [UW]

a. Find a function $$m=f(s)$$ that converts seconds $$s$$ into minutes $$m$$. Write out the formula for the new function $$C(f(s))$$; what does this function calculate?
b. Find a function $$m=g(h$$) that converts hours $$h$$ into minutes $$m$$. Write out the formula for the new function $$C(g(h))$$; what does this function calculate?
c. Find a function $$z=v(s)$$ that converts mph $$s$$ into ft/sec $$z$$. Write out the formula for the new function $$v(C(m)$$; what does this function calculate?

1. $$f(g(0)) = 36$$. $$g(f(0)) = -57$$

3. $$f(g(0)) = 4$$. $$g(f(0)) = 4$$

5. 4

7. 9

11. 7

13. 0

15. 4

17. 3

19. 2

21. $$f(g(x)) = \dfrac{x}{7}$$ $$g(f(x)) = 7x - 36$$

23. $$f(g(x)) = x + 3$$ $$g(f(x)) = \sqrt{x^2 + 3}$$

25. $$f(g(x)) = |5x + 1|$$ $$g(f(x)) = 5|x| + 1$$

27. $$f(g(h(x))) = (\sqrt{x} - 6)^4 + 6$$

29. b

31. a. $$r(V(t)) = \sqrt{\dfrac{3(10 + 20t)}{4\pi}}$$
b. 4.609 in

33. $$(0, \infty)$$

35. $$(-\infty, \dfrac{1}{3}) \cup (\dfrac{1}{3}, 1) \cup (1, \infty)$$

37. $$[2, 5) \cup (5, \infty)$$

39. $$g(x) = x + 2$$, $$f(x) = x^2$$

41. $$f(x) = \dfrac{3}{x}$$, $$g(x) = x - 5$$

43. $$f(x) = 3 + \sqrt{x}$$, $$g(x) = x - 2$$, or $$f(x) = 3 + x$$, $$g(x) = \sqrt{x - 2}$$

45. a. $$f(f(x)) = a(ax + b) + b = (a^2)x + (ab + b)$$
b. $$g(x) = \sqrt{6} x - \dfrac{8}{\sqrt{6} + 1}$$ or $$g(x) = -\sqrt{6} x - \dfrac{8}{1 - \sqrt{6}}$$

47. a. $$C(f(s)) = \dfrac{70(\dfrac{s}{60})^2}{10 + (\dfrac{s}{60})^2}$$
b. $$C(g(h)) = \dfrac{70(60h)^2}{10 + (60h)^2}$$
c. $$v(C(m)) = \dfrac{5280}{3600} (\dfrac{70m^2}{10 + m^2})$$