Free ACT Math: Functions Questions and Answers 3 — Questions and Answers
Question 1: The graph of f(x) is shifted 3 units to the right and 2 units down. If f(x) = x², what is the equation of the new function?
- g(x) = (x − 3)² − 2 (Correct answer)
- g(x) = (x + 3)² − 2
- g(x) = (x − 3)² + 2
- g(x) = x² − 5
Correct answer: g(x) = (x − 3)² − 2
Horizontal shift right by 3: replace x with (x − 3). Vertical shift down by 2: subtract 2. Result: g(x) = (x − 3)² − 2.
Question 2: For which value of x is the function f(x) = (x + 1)/(x² − x − 6) undefined?
- x = 3 and x = −2 (Correct answer)
- x = −1
- x = 3 only
- x = 6
Correct answer: x = 3 and x = −2
f(x) is undefined when the denominator equals zero. x² − x − 6 = (x − 3)(x + 2) = 0 → x = 3 or x = −2. Note: x = −1 makes the numerator zero but not the denominator, so it is defined (value = 0).
Question 3: A function f has the property f(−x) = −f(x) for all x. What type of function is f?
- An odd function (Correct answer)
- An even function
- A linear function
- A constant function
Correct answer: An odd function
A function is odd if f(−x) = −f(x) for all x. Odd functions have 180° rotational symmetry about the origin. Even functions satisfy f(−x) = f(x).
Question 4: If p(x) = 2x³ − x + 4, what is p(−1)?
- 3 (Correct answer)
- 5
- −1
- 7
Correct answer: 3
p(−1) = 2(−1)³ − (−1) + 4 = 2(−1) + 1 + 4 = −2 + 1 + 4 = 3.
Question 5: If f(x) = 5x + 2 and g(x) = x − 4, what is (f − g)(x)?
- 4x + 6 (Correct answer)
- 6x − 2
- 4x − 2
- 6x + 6
Correct answer: 4x + 6
(f − g)(x) = f(x) − g(x) = (5x + 2) − (x − 4) = 5x + 2 − x + 4 = 4x + 6.
Question 6: A function f is linear with f(2) = 7 and f(5) = 16. What is f(0)?
- 1 (Correct answer)
- 3
- 0
- −1
Correct answer: 1
Slope = (16 − 7)/(5 − 2) = 9/3 = 3. Using point-slope: f(x) = 3x + b. f(2) = 7: 6 + b = 7 → b = 1. So f(x) = 3x + 1. f(0) = 1.
Free ACT Math: Functions Questions and Answers 3
The graph of f(x) is shifted 3 units to the right and 2 units down.
If f(x) = x², what is the equation of the new function?