Algebra 2 Practice Test: Function Composition
- Function Composition
- 10 questions
- 15 minutes
These 10 questions practice evaluating and writing composite functions, using tables, finding the domain of a composition and splitting a function into two parts. In f(g(x)) the inner function g acts first, and its output becomes the input of f.
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Question 1
f(x) = 2x + 3
g(x) = x2 − 1
What is f(g(2))?
Correct answer: B
g(2) = 22 − 1 = 3, then f(3) = 2 · 3 + 3 = 9. Computing g(f(2)) = g(7) = 48 reverses the order.
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Question 2
f(x) = 2x + 3
g(x) = x2 − 1
What is g(f(2))?
Correct answer: A
f(2) = 2 · 2 + 3 = 7, then g(7) = 72 − 1 = 48. Forgetting to subtract 1 gives 49.
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Question 3
f(x) = 2x + 3
g(x) = x2 − 1
Which expression is (f ∘ g)(x)?
Correct answer: D
(f ∘ g)(x) = f(g(x)) = 2(x2 − 1) + 3 = 2x2 − 2 + 3 = 2x2 + 1. The expression 4x2 + 12x + 8 is g(f(x)), and the cubic is the product f · g.
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Question 4
f(x) = 2x + 3
g(x) = x2 − 1
Which expression is (g ∘ f)(x)?
Correct answer: A
(g ∘ f)(x) = (2x + 3)2 − 1 = 4x2 + 12x + 9 − 1 = 4x2 + 12x + 8. Squaring 2x + 3 as 4x2 + 9 drops the middle term 12x.
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Question 5
The function h(x) = 3x − 5 is written as h(x) = f(g(x)). Which choice works?
Correct answer: A
The inner function g is done first: compute 3x − 5, then take the square root with f. Swapping them gives 3x − 5, and 3x composed with x − 5 gives 3x − 15.
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Question 6
Let f(x) = x and g(x) = x − 4. What is the domain of f(g(x))?
Correct answer: B
f(g(x)) = x − 4 is defined only when x − 4 ≥ 0, so x ≥ 4. The domain of f alone, x ≥ 0, applies to the output of g, not to x.
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Question 7
x 1 2 3 4 f(x) 3 1 4 2 g(x) 4 1 2 3 Using the table, what is f(g(4))?
Correct answer: C
First g(4) = 3, then f(3) = 4. Reversing the order gives g(f(4)) = g(2) = 1, and reading only f(4) gives 2.
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Question 8
Let f(x) = x + 5. Which function g makes f(g(x)) = 3x − 1?
Correct answer: C
f(g(x)) = g(x) + 5, so g(x) + 5 = 3x − 1 and g(x) = 3x − 6. Adding 5 instead of subtracting it gives 3x + 4.
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Question 9
Let f(x) = 3x − 2. Which expression is f(f(x))?
Correct answer: B
f(f(x)) = 3(3x − 2) − 2 = 9x − 6 − 2 = 9x − 8. The quadratic is the product f(x) · f(x), not the composition.
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Question 10
A print shop makes n(t) = 20t shirts in t hours, and the cost of making n shirts is c(n) = 12n + 50 dollars. What is c(n(3))?
Correct answer: D
n(3) = 20 · 3 = 60 shirts, and c(60) = 12 · 60 + 50 = $770, the cost of the shirts made in 3 hours. c(3) = 86 uses the hours as if they were shirts.
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