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Study Guides > College Algebra

Section Exercises

1. Describe why the horizontal line test is an effective way to determine whether a function is one-to-one? 2. Why do we restrict the domain of the function f(x)=x2f\left(x\right)={x}^{2} to find the function’s inverse? 3. Can a function be its own inverse? Explain. 4. Are one-to-one functions either always increasing or always decreasing? Why or why not? 5. How do you find the inverse of a function algebraically? 6. Show that the function f(x)=a−xf\left(x\right)=a-x is its own inverse for all real numbers aa. For the following exercises, find f−1(x){f}^{-1}\left(x\right) for each function. 7. f(x)=x+3f\left(x\right)=x+3 8. f(x)=x+5f\left(x\right)=x+5 9. f(x)=2−xf\left(x\right)=2-x 10. f(x)=3−xf\left(x\right)=3-x 11. f(x)=xx+2f\left(x\right)=\frac{x}{x+2}\\ 12. f(x)=2x+35x+4f\left(x\right)=\frac{2x+3}{5x+4} For the following exercises, find a domain on which each function ff is one-to-one and non-decreasing. Write the domain in interval notation. Then find the inverse of ff restricted to that domain. 13. f(x)=(x+7)2f\left(x\right)={\left(x+7\right)}^{2} 14. f(x)=(x−6)2f\left(x\right)={\left(x - 6\right)}^{2} 15. f(x)=x2−5f\left(x\right)={x}^{2}-5 16. Given f(x)=x2+xf\left(x\right)=\frac{x}{2}+x\\ and g(x)=2x1−xg\left(x\right)=\frac{2x}{1-x}\\ a. Find f(g(x))f\left(g\left(x\right)\right) and g(f(x))g\left(f\left(x\right)\right)\\ b. What does the answer tell us about the relationship between f(x)f\left(x\right) and g(x)?g\left(x\right)? For the following exercises, use function composition to verify that f(x)f\left(x\right) and g(x)g\left(x\right) are inverse functions. 17. f(x)=x−13f\left(x\right)=\sqrt[3]{x - 1} and g(x)=x3+1g\left(x\right)={x}^{3}+1 18. f(x)=−3x+5f\left(x\right)=-3x+5 and g(x)=x−5−3g\left(x\right)=\frac{x - 5}{-3} For the following exercises, use a graphing utility to determine whether each function is one-to-one. 19. f(x)=xf\left(x\right)=\sqrt{x} 20. f(x)=3x+13f\left(x\right)=\sqrt[3]{3x+1} 21. f(x)=−5x+1f\left(x\right)=-5x+1 22. f(x)=x3−27f\left(x\right)={x}^{3}-27 For the following exercises, determine whether the graph represents a one-to-one function. 23. Graph of an upright parabola with vertex at (10, -10), passing through (0,0) and (25,0) 24. Graph of a step-function, with y = -5 for {x|-10<=x,0} and y = 0 for {x|0<=x<10} For the following exercises, use the graph of ff shown in [link]. Graph of the line y = (-3/2)x + 3 25. Find f(0)f\left(0\right). 26. Solve f(x)=0f\left(x\right)=0. 27. Find f−1(0){f}^{-1}\left(0\right). 28. Solve f−1(x)=0{f}^{-1}\left(x\right)=0. For the following exercises, use the graph of the one-to-one function shown below. Graph of a square root function for {x|x>=2} 29. Sketch the graph of f−1{f}^{-1}. 30. Find f(6) and f−1(2)f\left(6\right)\text{ and }{f}^{-1}\left(2\right). 31. If the complete graph of ff is shown, find the domain of ff. 32. If the complete graph of ff is shown, find the range of ff. For the following exercises, evaluate or solve, assuming that the function ff is one-to-one. 33. If f(6)=7f\left(6\right)=7, find f−1(7){f}^{-1}\left(7\right). 34. If f(3)=2f\left(3\right)=2, find f−1(2){f}^{-1}\left(2\right). 35. If f−1(−4)=−8{f}^{-1}\left(-4\right)=-8, find f(−8)f\left(-8\right). 36. If f−1(−2)=−1{f}^{-1}\left(-2\right)=-1, find f(−1)f\left(-1\right). For the following exercises, use the values listed in the table below to evaluate or solve.
xx f(x)f\left(x\right)
0 8
1 0
2 7
3 4
4 2
5 6
6 5
7 3
8 9
9 1
37. Find f(1)f\left(1\right). 38. Solve f(x)=3f\left(x\right)=3. 39. Find f−1(0){f}^{-1}\left(0\right). 40. Solve f−1(x)=7{f}^{-1}\left(x\right)=7. 41. Use the tabular representation of ff to create a table for f−1(x){f}^{-1}\left(x\right).
xx 3 6 9 13 14
f(x)f\left(x\right) 1 4 7 12 16
For the following exercises, find the inverse function. Then, graph the function and its inverse. 42. f(x)=3x−2f\left(x\right)=\frac{3}{x - 2} 43. f(x)=x3−1f\left(x\right)={x}^{3}-1 44. Find the inverse function of f(x)=1x−1f\left(x\right)=\frac{1}{x - 1}. Use a graphing utility to find its domain and range. Write the domain and range in interval notation. 45. To convert from xx degrees Celsius to yy degrees Fahrenheit, we use the formula f(x)=95x+32f\left(x\right)=\frac{9}{5}x+32. Find the inverse function, if it exists, and explain its meaning. 46. The circumference CC of a circle is a function of its radius given by C(r)=2πrC\left(r\right)=2\pi r. Express the radius of a circle as a function of its circumference. Call this function r(C)r\left(C\right). Find r(36π)r\left(36\pi \right) and interpret its meaning. 47. A car travels at a constant speed of 50 miles per hour. The distance the car travels in miles is a function of time, tt, in hours given by d(t)=50td\left(t\right)=50t. Find the inverse function by expressing the time of travel in terms of the distance traveled. Call this function t(d)t\left(d\right). Find t(180)t\left(180\right) and interpret its meaning.

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