Rucete ✏ AP Precalculus In a Nutshell
4. Parent Functions — Practice Questions 2
This chapter explores the graphs, transformations, inverses, and applications of parent functions.
(Multiple Choice — Click to Reveal Answer)
1. Which of the following is the parent function of all linear functions?
(A) f(x) = 2x + 3
(B) f(x) = x
(C) f(x) = x²
(D) f(x) = |x|
Answer
(B) f(x) = x. It is the simplest linear function passing through the origin with slope 1.
2. What is the domain of the square root parent function f(x) = √x?
(A) x ≥ 0
(B) x ≤ 0
(C) all real numbers
(D) x > −1
Answer
(A) x ≥ 0. Square roots are only defined for nonnegative inputs in real numbers.
3. The cubic parent function f(x) = x³ is symmetric about:
(A) the x-axis
(B) the y-axis
(C) the origin
(D) the line y = x
Answer
(C) the origin. Cubic functions are odd functions.
4. Which transformation results from f(x) = (x − 4)² compared to f(x) = x²?
(A) shift left 4
(B) shift right 4
(C) shift up 4
(D) shift down 4
Answer
(B) shift right 4. Subtracting inside the function shifts it right.
5. What is the range of the absolute value parent function f(x) = |x|?
(A) (−∞, ∞)
(B) [0, ∞)
(C) (−∞, 0]
(D) (0, ∞)
Answer
(B) [0, ∞). Absolute values are never negative.
6. Which of the following represents a vertical stretch of f(x) = x² by a factor of 3?
(A) f(x) = (x/3)²
(B) f(x) = 3x²
(C) f(x) = x² + 3
(D) f(x) = √x
Answer
(B) f(x) = 3x². Multiplying outside scales vertically.
7. Which of the following is NOT a parent function?
(A) f(x) = x²
(B) f(x) = 2x + 1
(C) f(x) = |x|
(D) f(x) = √x
Answer
(B) f(x) = 2x + 1. The parent linear function is f(x) = x, not a translated version.
8. The inverse of f(x) = 3x + 2 is:
(A) f⁻¹(x) = (x − 2)/3
(B) f⁻¹(x) = 3x − 2
(C) f⁻¹(x) = (x + 2)/3
(D) f⁻¹(x) = 1/(3x + 2)
Answer
(A) (x − 2)/3. Swap x and y, solve for y.
9. Which of the following is a piecewise-defined parent function?
(A) f(x) = x²
(B) f(x) = |x|
(C) f(x) = 1/x
(D) f(x) = x³
Answer
(B) |x|. Defined differently for x ≥ 0 and x < 0.
10. Which parent function has a horizontal asymptote at y = 0?
(A) f(x) = √x
(B) f(x) = x²
(C) f(x) = 1/x
(D) f(x) = |x|
Answer
(C) f(x) = 1/x. Rational functions approach zero as |x| → ∞.
11. What is the y-intercept of the quadratic parent function f(x) = x²?
(A) (0, 0)
(B) (1, 1)
(C) (0, 1)
(D) (−1, 1)
Answer
(A) (0, 0). Substituting x = 0 gives f(0) = 0.
12. Which parent function is decreasing on (0, ∞)?
(A) f(x) = |x|
(B) f(x) = 1/x
(C) f(x) = x²
(D) f(x) = x³
Answer
(B) f(x) = 1/x. Reciprocal decreases for positive x.
13. Which transformation results from f(x) = |x| − 5 compared to f(x) = |x|?
(A) Shift left 5
(B) Shift right 5
(C) Shift up 5
(D) Shift down 5
Answer
(D) Shift down 5. Subtracting outside moves the graph down.
14. The domain of f(x) = 1/(x−3) is:
(A) x ≠ 3
(B) all real numbers
(C) x > 3
(D) x < 3
Answer
(A) All real numbers except x = 3, since denominator cannot be zero.
15. The range of the square root function f(x) = √x is:
(A) (−∞, ∞)
(B) [0, ∞)
(C) (0, ∞)
(D) [−∞, 0]
Answer
(B) [0, ∞). Square roots are nonnegative.
16. Which parent function has a vertical asymptote at x = 0?
(A) f(x) = x²
(B) f(x) = 1/x
(C) f(x) = |x|
(D) f(x) = √x
Answer
(B) Reciprocal function has a vertical asymptote at x=0.
17. If g(x) = (x + 2)³, how is it transformed from f(x) = x³?
(A) Left 2
(B) Right 2
(C) Up 2
(D) Down 2
Answer
(A) Shift left 2. Adding inside shifts left.
18. Which parent function is odd?
(A) f(x) = |x|
(B) f(x) = x²
(C) f(x) = x³
(D) f(x) = √x
Answer
(C) f(x) = x³ is symmetric about the origin, so it is odd.
19. What is the inverse of f(x) = x³?
(A) f⁻¹(x) = √x
(B) f⁻¹(x) = x³
(C) f⁻¹(x) = ∛x
(D) f⁻¹(x) = x²
Answer
(C) The cube root function is the inverse of the cubic.
20. Which of these describes the graph of f(x) = −|x|?
(A) Opens upward, vertex at origin
(B) Opens downward, vertex at origin
(C) Opens upward, vertex at (0, −1)
(D) Opens downward, vertex at (0, −1)
Answer
(B) Reflection across x-axis flips the V-shape downward.
21. Which function has no inverse unless its domain is restricted?
(A) f(x) = x³
(B) f(x) = x²
(C) f(x) = √x
(D) f(x) = 1/x
Answer
(B) f(x) = x² fails horizontal line test unless restricted.
22. Which parent function models exponential growth?
(A) f(x) = 2^x
(B) f(x) = x²
(C) f(x) = |x|
(D) f(x) = 1/x
Answer
(A) Exponential parent function shows growth as x increases.
23. The line y = x is the axis of symmetry for the graph of:
(A) f(x) = |x|
(B) f(x) = 1/x
(C) f(x) = √x and f⁻¹(x) = x², x ≥ 0
(D) f(x) = x³
Answer
(C) Functions and their inverses are symmetric across y = x.
24. Which is the parent function of logarithms?
(A) f(x) = ln(x)
(B) f(x) = e^x
(C) f(x) = 10^x
(D) f(x) = log₁₀(x)
Answer
(A) Natural logarithm ln(x) is the parent log function.
25. Which parent function’s graph passes through (0,1)?
(A) f(x) = e^x
(B) f(x) = |x|
(C) f(x) = x²
(D) f(x) = √x
Answer
(A) f(x) = e^x. All exponential functions of base e pass through (0,1).
26. If f(x) = √(x−1), which restriction on the domain ensures the function is defined?
(A) x ≥ 1
(B) x > 0
(C) x ≠ 1
(D) all real x
Answer
(A) The radicand (x−1) must be nonnegative, so x ≥ 1.
27. Which transformation is applied to f(x) = x² to obtain g(x) = −2(x−3)² + 4?
(A) Right 3, vertical stretch by 2, reflect over x-axis, up 4
(B) Left 3, vertical stretch by 2, reflect over x-axis, up 4
(C) Right 3, vertical shrink by 1/2, reflect over y-axis, up 4
(D) Right 3, vertical stretch by 2, reflect over y-axis, up 4
Answer
(A) Inside (x−3) shifts right 3, −2 causes reflection and vertical stretch, +4 shifts up.
28. If f(x) = 2^x, what is the inverse function?
(A) f⁻¹(x) = 2^x
(B) f⁻¹(x) = log₂(x)
(C) f⁻¹(x) = ln(x)
(D) f⁻¹(x) = √x
Answer
(B) The inverse of an exponential function base 2 is log base 2.
29. A parabola with vertex (2, −3) opens upward. Which equation represents this?
(A) f(x) = (x−2)² − 3
(B) f(x) = −(x+2)² − 3
(C) f(x) = (x+2)² + 3
(D) f(x) = −(x−2)² + 3
Answer
(A) Vertex form: (x−h)² + k with h=2, k=−3.
30. Which composite is equal to x for all x in the domain?
(A) f(g(x)) if f and g are both even
(B) f(f(x)) if f is odd
(C) f(g(x)) if g is inverse of f
(D) g(g(x)) if g is inverse of f
Answer
(C) By definition, composition of inverse functions yields x.
31. If h(x) = |x−4| + 2, which describes its vertex and orientation?
(A) Vertex (−4, 2), opens up
(B) Vertex (4, 2), opens up
(C) Vertex (4, −2), opens up
(D) Vertex (−4, −2), opens up
Answer
(B) The absolute value shifts right 4 and up 2; opens upward as V-shape.
32. Which function has range (−∞, ∞)?
(A) f(x) = x²
(B) f(x) = |x|
(C) f(x) = x³
(D) f(x) = √x
Answer
(C) The cubic parent covers all real outputs.
33. If f(x) = log(x), what is its domain?
(A) (−∞, ∞)
(B) (0, ∞)
(C) [0, ∞)
(D) (−∞, 0)
Answer
(B) The logarithmic function is defined only for positive x.
34. The axis of symmetry of f(x) = (x+1)(x−3) is:
(A) x = 2
(B) x = −2
(C) x = 1
(D) x = −1
Answer
(C) Axis lies midway between roots −1 and 3, so x = 1.
35. Which parent function is neither even nor odd?
(A) f(x) = x²
(B) f(x) = x³
(C) f(x) = e^x
(D) f(x) = |x|
Answer
(C) Exponential functions are neither symmetric about y-axis nor origin.
36. Find the domain of f(x) = √(x+5).
Answer
Domain: x ≥ −5, since the radicand must be nonnegative.
37. Find the inverse of f(x) = (x−2)³.
Answer
Swap x and y: x = (y−2)³ → y = ∛x + 2. So f⁻¹(x) = ∛x + 2.
38. Find the x-intercepts of f(x) = (x−1)(x+3).
Answer
Set f(x)=0 → x=1 or x=−3. Intercepts: (1,0), (−3,0).
39. Determine the vertex of f(x) = (x+4)² − 7.
Answer
Vertex: (−4, −7), from vertex form.
40. State the range of f(x) = |x| − 2.
Answer
Range: [−2, ∞), since the minimum value is −2.
41. Solve for x: f(x) = 2^x = 16.
Answer
2^x = 16 → x=4.
42. Evaluate f(−2) if f(x) = 3x² − 5x + 1.
Answer
f(−2) = 3(4) − (−10) + 1 = 12+10+1 = 23.
43. Find the line of symmetry for f(x) = x² + 6x + 5.
Answer
Axis: x = −b/(2a) = −6/(2) = −3.
44. Determine the maximum value of h(t) = −t² + 6t + 5.
Answer
Vertex at t = −b/(2a) = −6/(−2) = 3. h(3)= −9+18+5=14. Maximum =14.
45. Find the y-intercept of f(x) = (x−2)(x+1).
Answer
Substitute x=0 → f(0) = (−2)(1)= −2. So (0,−2).
46. State whether f(x) = x³ − 4x is even, odd, or neither.
Answer
Odd. f(−x)= −x³+4x= −(x³−4x)= −f(x).
47. Write the inverse of f(x) = (x+1)/(x−2).
Answer
Swap: x=(y+1)/(y−2). Solve: x(y−2)=y+1 → xy−2x=y+1 → xy−y=2x+1 → y(x−1)=2x+1 → y=(2x+1)/(x−1).
48. Find the zeros of f(x) = x² − 9.
Answer
Set f(x)=0 → x²=9 → x=±3.
49. If g(x)=e^x, find g(0) and g(1).
Answer
g(0)=e⁰=1; g(1)=e≈2.718.
50. A function f(x)=√(x−4) is shifted 3 units left. Write the new equation.
Answer
Shift left: replace x with x+3 → f(x)=√((x+3)−4)=√(x−1).