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ACT · Math · Preparing for Higher Math: Functions
Nonlinear functions is part of Preparing for Higher Math: Functions in the ACT Math section. The questions below are written in the style of the real test, each with the answer and a worked explanation.
Question types you will see here include composite functions, domain and range restrictions, transformations of a graph and exponential growth or decay model.
Common traps: applying the outer function first; substituting into the wrong variable; multiplying the functions.
Question 1
A coffee shop in Melbourne prices a bag as f(x) = 2x − 6, where x is the weight in hundreds of grams. What is f⁻¹(10)?
2
4
8
14
Answer: C. 8
Solve 10 = 2x − 6: 2x = 16, so x = 8. A subtracts 6 instead of adding it when undoing; B subtracts 6 and forgets to divide by 2; D evaluates f(10) instead of the inverse.
Question 2
The number of cells in a culture is 2,400 cells and increases by 5% each year. Which expression gives the number of cells in a culture, in cells, t years from now?
2,400(0.05)^t
2,400(1 + 5t)
2,400(0.95)^t
2,400(1.05)^t
Answer: D. 2,400(1.05)^t
A quantity that increases by 5% each year is multiplied by 1 + 0.05 = 1.05 each year. After t years the amount is 2,400(1.05)^t.
Question 3
The function f is defined by f(x) = (2x + 1)/(x − 3) for x ≠ 3. What is f⁻¹(3)?
−8
1
10
21
Answer: C. 10
C is right: Solve (2x + 1)/(x − 3) = 3: 2x + 1 = 3x − 9, so x = 10. A makes a sign slip when distributing, 2x + 1 = 3x + 9. B sets only the numerator equal to 3. D stops at 2x + 1 = 21 and reports that instead of x.
Question 4
The function f is defined by f(x) = (x² − 9)/(x² − x − 6). At which x-values does the graph of f have a vertical asymptote?
x = 3 only
x = −2 and x = 3
x = −2 only
x = −3 and x = 3
Answer: C. x = −2 only
Factor: f(x) = (x − 3)(x + 3)/((x − 3)(x + 2)). The factor x − 3 cancels, so x = 3 is a hole in the graph, not an asymptote, and only x = −2 remains. Choice A keeps the hole and drops the asymptote; B counts both zeros of the denominator without canceling; D uses the zeros of the numerator.
Question 5
The graph of y = |x| is shifted 4 units to the right and 3 units down, and then the result is reflected across the x-axis. Which equation gives the final graph?
y = −|x − 4| + 3
y = −|x − 4| − 3
y = −|x + 4| + 3
y = −|x − 3| + 4
Answer: A. y = −|x − 4| + 3
Shift: y = |x − 4| − 3. Reflect across the x-axis by negating the whole expression: y = −(|x − 4| − 3) = −|x − 4| + 3. Choice B (y = −|x − 4| − 3) reflects first and then shifts down. Choice C (y = −|x + 4| + 3) shifts left instead of right. Choice D (y = −|x − 3| + 4) swaps the horizontal and vertical shifts.
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