Drill 1 · Math · Functions
ACT Math: Functions (Drill 1) is a Math practice drill covering Functions. It contains 5 original questions created by Brian Stewart, a Barron's test prep author with over 20 years of tutoring experience.
ACT Functions questions cover evaluating functions for given inputs, determining domain and range, reading functions defined by tables and graphs, composing two functions, and interpreting the effect of vertical and horizontal transformations on a graph.
Question 1. If f(x) = 2x2 − 3x + 1, what is f(−2)?
Explanation: Substitute x = −2: f(−2) = 2(−2)2 − 3(−2) + 1 = 2(4) + 6 + 1 = 8 + 6 + 1 = 15. Choice A (−1) results from treating (−2)2 as −4 instead of +4: 2(−4) + 6 + 1 = −8 + 7 = −1. Choice B (3) results from treating −3(−2) as −6 instead of +6: 8 − 6 + 1 = 3. Choice C (11) results from a combination of arithmetic errors across multiple steps; there is no single clean path to 11, which makes it a useful reminder to evaluate each term carefully and check the final sum.
Question 2. What is the domain of the function f(x) = (x + 4) / (x2 − x − 6)?
Explanation: The domain excludes values that make the denominator zero. Factor the denominator: x2 − x − 6 = (x − 3)(x + 2). Set each factor equal to zero: x = 3 and x = −2. The domain is all real numbers except x = 3 and x = −2. The numerator x + 4 does not affect the domain, x = −4 makes the numerator zero but not the denominator, so f(−4) = 0, which is defined. Choice A incorrectly excludes the zero of the numerator. Choice C results from factoring the denominator with wrong signs: (x + 3)(x − 2). Choice D ignores the restriction entirely.
Question 3. The table below shows values of a linear function g(x).
x −2 0 2 4
g(x) 1 5 9 13
Which of the following defines g(x)?
Explanation: The slope is the change in g(x) divided by the change in x. From x = 0 to x = 2: slope = (9 − 5) / (2 − 0) = 4/2 = 2. The y-intercept is g(0) = 5. So g(x) = 2x + 5. Verify: g(−2) = 2(−2) + 5 = 1 ✓; g(4) = 2(4) + 5 = 13 ✓. Choice B uses slope 3, which gives g(2) = 11 ≠ 9. Choice C uses slope 4, which gives a y-intercept of 1, check: g(0) = 1 ≠ 5. Choice D uses the correct slope of 2 but takes the y-intercept from g(−2) = 1 rather than g(0) = 5.
Question 4. Let f(x) = 3x − 2 and g(x) = x2 + 1. What is f(g(3))?
Explanation: Work from the inside out. First evaluate g(3): g(3) = 32 + 1 = 9 + 1 = 10. Then evaluate f(10): f(10) = 3(10) − 2 = 30 − 2 = 28. Choice A (26) results from correctly computing g(3) = 10 but then subtracting 2 twice in f: 3(10) − 2 − 2 = 26. Choice C (50) results from reversing the order of composition, computing g(f(3)) instead of f(g(3)): f(3) = 3(3) − 2 = 7, then g(7) = 72 + 1 = 50. Choice D (98) results from reversing the composition and then misapplying the coefficient in g: computing 2(72) = 98 rather than 72 + 1.
Question 5. The graph of f(x) = x2 is transformed to produce the graph of g(x) = (x − 3)2 + 4. Compared to the graph of f(x), the graph of g(x) is shifted in which direction?
Explanation: In vertex form g(x) = (x − h)2 + k, h is the horizontal shift and k is the vertical shift. Here h = 3 and k = 4. A positive h means the graph shifts right (the vertex moves from (0, 0) to (3, 4)). A positive k means the graph shifts up. So the graph shifts 3 units right and 4 units up. Choice A reverses the horizontal direction, the common error of reading (x − 3) as a leftward shift because of the minus sign. Choice B gets the horizontal direction right but flips the vertical direction. Choice D reverses both directions.