Free ACT Math: Functions Questions and Answers 1 — Questions and Answers
Question 1: If f(x) = 3x² - 5x + 2, what is the value of f(-3)?
- -40
- -10
- 14
- 44 (Correct answer)
Correct answer: 44
To find f(-3), substitute -3 for every instance of x in the function f(x) = 3x² - 5x + 2. The calculation is f(-3) = 3(-3)² - 5(-3) + 2 = 3(9) + 15 + 2 = 27 + 15 + 2 = 44. Be careful with the signs, especially when squaring a negative number.
Question 2: Let f(x) = x + 5 and g(x) = 2x - 3. What is f(g(x))?
- 2x + 2 (Correct answer)
- 2x + 7
- 2x² + 7x - 15
- 3x + 2
Correct answer: 2x + 2
The composition f(g(x)) means you substitute the entire function g(x) into the function f(x) wherever you see x. So, f(g(x)) = f(2x - 3). Now, substitute (2x - 3) into f(x) = x + 5, which gives (2x - 3) + 5. Simplifying this expression results in 2x + 2.
Question 3: Given f(x) = x² + 1 and g(x) = x - 2, what is the value of g(f(3))?
- 2
- 8 (Correct answer)
- 10
- 14
Correct answer: 8
This is a composition of functions evaluated at a point. First, evaluate the inner function: f(3) = (3)² + 1 = 9 + 1 = 10. Then, use this result as the input for the outer function: g(10) = 10 - 2 = 8. Therefore, g(f(3)) = 8.
Question 4: The graph of y = f(x) is shown in the xy-plane. Which of the following is the graph of y = f(x) - 2?
- The graph of f(x) shifted 2 units to the right.
- The graph of f(x) shifted 2 units to the left.
- The graph of f(x) shifted 2 units up.
- The graph of f(x) shifted 2 units down. (Correct answer)
Correct answer: The graph of f(x) shifted 2 units down.
The transformation y = f(x) + k represents a vertical shift of the graph of f(x). When k is negative, the shift is downward. Therefore, y = f(x) - 2 is the graph of f(x) shifted down by 2 units.
Question 5: If h(x) is a function, how does the graph of y = h(x + 4) relate to the graph of y = h(x)?
- It is shifted 4 units to the left. (Correct answer)
- It is shifted 4 units to the right.
- It is shifted 4 units up.
- It is shifted 4 units down.
Correct answer: It is shifted 4 units to the left.
The transformation y = f(x + k) represents a horizontal shift. Contrary to intuition, a positive k (like +4) shifts the graph to the left, and a negative k shifts it to the right. Therefore, y = h(x + 4) is the graph of h(x) shifted 4 units to the left.
Question 6: The function f(x) = x² is transformed to g(x) = (x - 3)² + 5. Which statement describes this transformation?
- A shift 3 units to the left and 5 units down.
- A shift 3 units to the left and 5 units up.
- A shift 3 units to the right and 5 units up. (Correct answer)
- A shift 3 units to the right and 5 units down.
Correct answer: A shift 3 units to the right and 5 units up.
This transformation combines a horizontal and a vertical shift. The (x - 3) part indicates a horizontal shift 3 units to the right. The +5 part indicates a vertical shift 5 units up. Combining these gives a shift 3 units right and 5 units up.
Question 7: For the function f(x) = 2|x - 3|, what is the value of f(1)?
- -4
- -2
- 4 (Correct answer)
- 8
Correct answer: 4
To evaluate f(1), substitute 1 for x in the function. This gives f(1) = 2|1 - 3|. First, calculate the value inside the absolute value bars: 1 - 3 = -2. The absolute value of -2 is 2. Finally, multiply by the coefficient: 2 * 2 = 4.
Free ACT Math: Functions Questions and Answers 1
If f(x) = 3x² - 5x + 2, what is the value of f(-3)?