PRE CALCULUS Functions 2 — Questions and Answers
Question 1: Which of the following is the definition of a one-to-one function?
- Each output has exactly one input (Correct answer)
- Each input has exactly one output
- The function passes the vertical line test
- The domain equals the range
Correct answer: Each output has exactly one input
A one-to-one (injective) function means each output value corresponds to exactly one input value.
Question 2: If f(x) = 2x - 3, what is f⁻¹(x)?
- (x + 3) / 2 (Correct answer)
- (x - 3) / 2
- 2x + 3
- 1 / (2x - 3)
Correct answer: (x + 3) / 2
To find the inverse, swap x and y in y = 2x - 3 and solve for y: x = 2y - 3 → y = (x + 3)/2.
Question 3: Given f(x) = x² and g(x) = x + 1, what is (f ∘ g)(x)?
- x² + 1
- (x + 1)²
- x² + 2x + 1
- (x + 1)² and x² + 2x + 1 are the same (Correct answer)
Correct answer: (x + 1)² and x² + 2x + 1 are the same
(f ∘ g)(x) = f(g(x)) = (x+1)² = x² + 2x + 1, so options B and C are identical expressions.
Question 4: What is the range of f(x) = |x| - 4?
- y ≥ -4 (Correct answer)
- y ≤ -4
- All real numbers
- y ≥ 4
Correct answer: y ≥ -4
Since |x| ≥ 0 for all x, |x| - 4 ≥ -4, so the range is [-4, ∞).
Question 5: Which transformation maps f(x) to f(x - 3) + 2?
- Shift left 3, up 2
- Shift right 3, up 2 (Correct answer)
- Shift right 3, down 2
- Shift left 3, down 2
Correct answer: Shift right 3, up 2
Replacing x with (x - 3) shifts the graph right 3 units, and adding 2 shifts it up 2 units.
Question 6: A function f is even if:
- f(-x) = f(x) for all x (Correct answer)
- f(-x) = -f(x) for all x
- f(x) = f(x + 2) for all x
- The graph passes the vertical line test
Correct answer: f(-x) = f(x) for all x
An even function satisfies f(-x) = f(x), meaning its graph is symmetric about the y-axis.
Question 7: If f(x) = √(x - 5), what is the domain of f?
- x ≥ 5 (Correct answer)
- x > 5
- x ≤ 5
- All real numbers
Correct answer: x ≥ 5
The expression under the square root must be non-negative: x - 5 ≥ 0, so x ≥ 5.
PRE CALCULUS Functions 2
Which of the following is the definition of a one-to-one function?