12.8.A.1Multiple Choice
Which function is invertible on the set of all real numbers?
Select A, B, C, or D to check your answer.
Directions: Work each problem on your own paper first. For multiple choice, select A, B, C, or D. You will see right away whether you are correct. If you miss, you will see the most likely reason, and you can try again. For short response, type your answer, then compare it with the model solution. Questions 1–16 go through 2.8.A.1 to 2.8.B.2 in order, and questions 17–20 use inverses in context and mix the ideas. A graphing calculator is allowed for solving. Round sensibly: money to the cent, other decimals to the nearest hundredth.
Which function is invertible on the set of all real numbers?
Select A, B, C, or D to check your answer.
Two functions are given by tables. (a) Which one is invertible? Explain. (b) For the other, give a domain with three inputs on which it is invertible.
| x | 1 | 2 | 3 | 4 |
| f(x) | 5 | 8 | 5 | 2 |
| x | 1 | 2 | 3 | 4 |
| g(x) | 9 | 6 | 2 | 4 |
Let f(x) = (x + 3)². On which domain is f invertible?
Select A, B, C, or D to check your answer.
The function f is invertible. (a) Find f⁻¹(9) and f⁻¹(1). (b) Find f(f⁻¹(6)). (c) Is f⁻¹(10) defined? Explain.
| x | 1 | 2 | 3 | 4 | 5 |
| f(x) | 14 | 9 | 6 | 2 | 1 |
The function f is invertible and f(4) = 11. Which statement must be true?
Select A, B, C, or D to check your answer.
A shipping company charges C(w) = 4 + 0.5w dollars to ship w pounds. (a) Find C(10). (b) Find C⁻¹(14). (c) Explain what C⁻¹(14) means. (d) What are the units of the input and output of C⁻¹?
The graph of an invertible function f passes through the point (2, 5). Which point must be on the graph of f⁻¹?
Select A, B, C, or D to check your answer.
Let f(x) = 2x + 5. (a) Find f⁻¹(11) by finding the input that gives output 11. (b) Find f⁻¹(−1). (c) Find f⁻¹(5). (d) Verify one answer with f.
Which statement shows that g is the inverse of f?
Select A, B, C, or D to check your answer.
Let f(x) = 3x − 6. (a) Show that g(x) = (x + 6)/3 is the inverse of f by composing in both orders. (b) Is h(x) = (x + 2)/3 the inverse of f? Show your work.
The inverse of f(x) = x³ is f⁻¹(x) = ∛x. What is f(f⁻¹(−8))?
Select A, B, C, or D to check your answer.
Let f(x) = 5 − 2x and g(x) = (5 − x)/2. (a) Find f(g(3)) and g(f(3)). (b) Show that f(g(x)) = x and g(f(x)) = x. (c) How does this connect to the identity function I(x) = x?
An invertible function f has domain x ≥ 3 and range y ≥ −2. What are the domain and range of f⁻¹?
Select A, B, C, or D to check your answer.
Let f(x) = √(x − 2) + 1 for x ≥ 2. (a) Give the domain and range of f. (b) Give the domain and range of f⁻¹. (c) Find a formula for f⁻¹. (d) Check by finding f(6) and then f⁻¹ of that value.
An invertible function f has the input-output pairs (1, 8), (2, 5), (3, 3), (4, 1). Which list gives the input-output pairs of f⁻¹?
Select A, B, C, or D to check your answer.
Let f(x) = (x − 1)³ + 4. (a) List the steps f performs on x. (b) Undo them in reverse order to find f⁻¹. (c) Check by finding f(3) and then f⁻¹ of that value. (d) Give the domain and range of f⁻¹.
The temperature in degrees Fahrenheit is F(c) = 1.8c + 32 when the temperature is c degrees Celsius. What does F⁻¹(77) represent?
Select A, B, C, or D to check your answer.
An object falls d(t) = 16t² feet in t seconds, for t ≥ 0. (a) Find d(3). (b) Explain why d is invertible on t ≥ 0. (c) Find a formula for d⁻¹. (d) How long does it take to fall 100 feet? (e) What are the units of the input and the output of d⁻¹?
Let f(x) = x² − 6x with domain x ≥ 3, on which f is invertible. What is f⁻¹(16)?
Select A, B, C, or D to check your answer.
Let f(x) = (x + 1)² − 3 with domain x ≥ −1. (a) Explain why f is invertible on this domain. (b) Give the range of f. (c) Find a formula for f⁻¹ and its domain. (d) Find f⁻¹(f(2)). (e) Show that f(f⁻¹(y)) = y.