12.7.A.1Multiple Choice
Which expression is equal to (f∘g)(x)?
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 1–19 go through 2.7.A.1 to 2.7.C.3 in order, and question 20 mixes them. A graphing calculator is allowed for regression and for solving. Round sensibly: money to the cent, counts of people or cells to the nearest whole number, other decimals to the nearest hundredth.
Which expression is equal to (f∘g)(x)?
Select A, B, C, or D to check your answer.
Let f(x) = √x and g(x) = 10 − 2x. (a) Find (f∘g)(3). (b) Find (f∘g)(8), or explain why it is undefined. (c) Find the domain of f∘g.
Let f(x) = 1/(x − 2) and g(x) = x + 6. Which input value is NOT in the domain of f∘g?
Select A, B, C, or D to check your answer.
Use the tables. (a) Find (f∘g)(0), (f∘g)(1), (f∘g)(2), and (f∘g)(3), or write “undefined.” (b) Find (g∘f)(3) and (g∘f)(4).
| x | 1 | 2 | 3 | 4 | 5 |
| f(x) | 6 | 9 | 0 | 3 | 8 |
| x | 0 | 1 | 2 | 3 | |
| g(x) | 2 | 5 | 1 | 7 |
Values of f and g are given. What is f(g(3))?
| x | 1 | 2 | 3 | 4 | 5 |
| f(x) | 3 | 5 | 2 | 1 | 6 |
| g(x) | 4 | 1 | 5 | 2 | — |
Select A, B, C, or D to check your answer.
A worker earns g(h) = 12h dollars in h hours before taxes. After taxes the worker keeps f(d) = 0.85d dollars from d dollars earned. (a) Find f(g(5)). (b) Write a formula for f(g(h)). (c) What does f(g(h)) represent?
Let f(x) = x² − 4x and g(x) = x + 1. What is (f∘g)(3)?
Select A, B, C, or D to check your answer.
Let f(x) = 2x + 1 and g(x) = x − 3. (a) Write f(g(x)) and g(f(x)) as simplified expressions. (b) Find both at x = 4. (c) Are f∘g and g∘f the same function?
Let f(x) = 2x + 1 and g(x) = x². What is (g∘f)(3)?
Select A, B, C, or D to check your answer.
A jacket costs $80. A sale gives 25% off, f(p) = 0.75p, and a coupon takes $10 off, g(p) = p − 10. (a) Find the price if the sale discount is applied first. (b) Find the price if the coupon is applied first. (c) Which order is cheaper, and by how much?
Let I(x) = x be the identity function and let g be any function. Which statement is always true?
Select A, B, C, or D to check your answer.
Let g(x) = 2x − 5 and I(x) = x. (a) Find (g∘I)(4) and (I∘g)(4). (b) Is h(x) = x + 2 an identity function? Test it with g(4). (c) Which number is the identity for addition, and which is the identity for multiplication?
A circular stain has radius r(t) = 4t centimeters after t minutes. Its area is A(r) = πr² square centimeters. Which gives the area as a function of time?
Select A, B, C, or D to check your answer.
Let f(x) = x² − x and g(x) = 3x + 2. (a) Write f(g(x)) as a simplified expression. (b) Write g(f(x)) as a simplified expression. (c) Use your formulas to find f(g(1)) and g(f(2)).
Let f(x) = √(x + 3) and g(x) = x² − 1. Which is f(g(x))?
Select A, B, C, or D to check your answer.
Use the tables. (a) Make a table of f∘g for the inputs of g. (b) For which inputs is g∘f defined, and what are its values?
| x | 2 | 4 | 6 | 8 |
| g(x) | 1 | 3 | 5 | 7 |
| x | 1 | 3 | 5 | 7 |
| f(x) | 10 | 4 | 9 | 16 |
The function h(x) = (5 − 2x)³ is written as f(g(x)). Which pair of functions f and g works?
Select A, B, C, or D to check your answer.
(a) Write h(x) = 1/(x² + 4) as f(g(x)), and check with x = 1. (b) Write k(x) = 2·5x+1 − 3 as a composition of three functions a, b, and c, and check with x = 1.
The graph of h(x) = f(3x) is the composition of g(x) = 3x with f. How does the graph of h compare with the graph of f?
Select A, B, C, or D to check your answer.
Let f(x) = 2x, g(x) = x + 3, and h(x) = 3x. (a) Write f(g(x)) and g(f(x)). (b) Find f(g(1)) and g(f(1)). (c) Write f(h(x)) in the form bx. (d) Describe the graph of f(g(x)) as a transformation of f. (e) Write m(x) = (2x + 3)² as a composition of three functions, and check with x = 2.