Free ACT Math: Functions Questions and Answers — Questions and Answers
Question 1: If f(x) = 3x² − 2x + 1, what is f(−2)?
- 17 (Correct answer)
- 9
- 5
- −3
Correct answer: 17
Substitute x = −2 into the function: f(−2) = 3(−2)² − 2(−2) + 1 = 3(4) + 4 + 1 = 12 + 4 + 1 = 17. Remember that (−2)² = +4 and −2(−2) = +4.
Question 2: Which of the following represents the inverse function of f(x) = 2x + 6?
- f⁻¹(x) = (x − 6)/2 (Correct answer)
- f⁻¹(x) = 2x − 6
- f⁻¹(x) = (x + 6)/2
- f⁻¹(x) = x/2 + 6
Correct answer: f⁻¹(x) = (x − 6)/2
To find the inverse, swap x and y, then solve for y. Starting with y = 2x + 6: swap to get x = 2y + 6, then solve: x − 6 = 2y, so y = (x − 6)/2. Therefore f⁻¹(x) = (x − 6)/2.
Question 3: If g(x) = x + 3 and h(x) = x², what is h(g(2))?
- 25 (Correct answer)
- 7
- 13
- 49
Correct answer: 25
This is a composition of functions. First evaluate the inner function: g(2) = 2 + 3 = 5. Then evaluate the outer function with this result: h(5) = 5² = 25. Always work from the inside out with function composition.
Question 4: A function f is defined as f(x) = |x − 3|. For what value(s) of x does f(x) = 5?
- x = 8 and x = −2 (Correct answer)
- x = 8 only
- x = −2 only
- x = 2 and x = 8
Correct answer: x = 8 and x = −2
Set up the absolute value equation: |x − 3| = 5. This gives two cases: x − 3 = 5 → x = 8, OR x − 3 = −5 → x = −2. Always check both cases when solving absolute value equations.
Question 5: The function f(x) = x² − 4x + 3 has zeros at x = 1 and x = 3. What is the x-coordinate of the vertex of this parabola?
- x = 2 (Correct answer)
- x = 1
- x = 3
- x = 4
Correct answer: x = 2
The vertex of a parabola lies on the axis of symmetry, which is the vertical line exactly halfway between the two zeros. Midpoint of x = 1 and x = 3: (1 + 3)/2 = 4/2 = 2. Alternatively, use the vertex formula x = −b/2a = −(−4)/(2·1) = 4/2 = 2.
Question 6: If f(x) = 2ˣ, which statement correctly describes the behavior of this function?
- The function is always positive and increases as x increases (Correct answer)
- The function crosses the x-axis at x = 2
- The function decreases as x increases
- The function has a maximum value of 2
Correct answer: The function is always positive and increases as x increases
f(x) = 2ˣ is an exponential growth function. Its key properties: always positive (range is y > 0), y-intercept at (0, 1) since 2⁰ = 1, increasing for all x, has a horizontal asymptote at y = 0 (but never equals 0), and domain is all real numbers.
Free ACT Math: Functions Questions and Answers
If f(x) = 3x² − 2x + 1, what is f(−2)?