Rucete ✏ AP Precalculus In a Nutshell
4. Parent Functions — Practice Questions 3
This chapter explores the graphs, transformations, inverses, and modeling applications of parent functions.
(Multiple Choice — Click to Reveal Answer)
1. Which of the following is the parent function of all quadratic functions?
A. f(x) = x + 1
B. f(x) = x²
C. f(x) = |x|
D. f(x) = √x
Answer
B. The parent quadratic is f(x)=x², all others are transformations or different families.
2. Which transformation shifts f(x)=|x| three units to the right?
A. f(x) = |x| + 3
B. f(x) = |x−3|
C. f(x) = |x+3|
D. f(x) = −|x|
Answer
B. Subtracting inside the absolute value shifts right.
3. The domain of f(x)=√(x−2) is:
A. x ≥ −2
B. x ≤ 2
C. x ≥ 2
D. All real x
Answer
C. Inside the square root must be nonnegative: x−2 ≥ 0 → x ≥ 2.
4. Which function is a vertical stretch of f(x)=x³ by factor of 4?
A. f(x)=4x³
B. f(x)=(x/4)³
C. f(x)=x³+4
D. f(x)=(x+4)³
Answer
A. Multiplying the output by 4 produces a vertical stretch.
5. Which graph represents an odd function?
A. y=x²
B. y=|x|
C. y=x³
D. y=√x
Answer
C. Cubic passes through origin and is symmetric about the origin, a hallmark of odd functions.
6. Which transformation reflects f(x)=x² across the x-axis?
A. f(x)=−x²
B. f(x)=(−x)²
C. f(x)=x²−1
D. f(x)=(x−1)²
Answer
A. Multiplying the output by −1 reflects across the x-axis.
7. Which parent function has domain x ≠ 0 and range y ≠ 0?
A. f(x)=1/x
B. f(x)=x²
C. f(x)=√x
D. f(x)=|x|
Answer
A. Reciprocal function is undefined at 0, and its output never equals 0.
8. What is the inverse of f(x)=x³?
A. f⁻¹(x)=x²
B. f⁻¹(x)=³√x
C. f⁻¹(x)=√x
D. f⁻¹(x)=1/x³
Answer
B. The cubic inverse is the cube root function.
9. Which of the following is even?
A. f(x)=x³
B. f(x)=x
C. f(x)=|x|
D. f(x)=√x
Answer
C. |x| is symmetric across the y-axis, making it even.
10. Which function has no maximum or minimum value?
A. f(x)=x²
B. f(x)=x³
C. f(x)=−x²
D. f(x)=√x
Answer
B. Cubic extends to infinity in both directions without bound.
11. Which is the effect of f(x)=x²+5?
A. Shift left 5
B. Shift right 5
C. Shift up 5
D. Shift down 5
Answer
C. Adding 5 outside shifts graph up.
12. The range of f(x)=√x is:
A. (−∞,∞)
B. [0,∞)
C. (−∞,0]
D. (−∞,∞) excluding 0
Answer
B. Square root outputs only nonnegative values.
13. Which function is compressed vertically by 1/2?
A. f(x)=0.5x²
B. f(x)=(2x)²
C. f(x)=x²+0.5
D. f(x)=(x+0.5)²
Answer
A. Multiplying output by 0.5 shrinks vertically.
14. Which has domain x ≥ −3?
A. f(x)=√(x+3)
B. f(x)=√(x−3)
C. f(x)=|x+3|
D. f(x)=1/(x+3)
Answer
A. Inside sqrt requires x+3 ≥ 0 → x ≥ −3.
15. What is f⁻¹(x) if f(x)=x+7?
A. f⁻¹(x)=x−7
B. f⁻¹(x)=7−x
C. f⁻¹(x)=1/x−7
D. f⁻¹(x)=x/7
Answer
A. Subtract 7 to undo the addition.
16. Which parent function’s graph passes through (0,0) and (1,1)?
A. f(x)=x²
B. f(x)=|x|
C. f(x)=x³
D. f(x)=√x
Answer
D. √x passes through both (0,0) and (1,1).
17. Which of these is not a parent function?
A. f(x)=x²
B. f(x)=x³
C. f(x)=√(x−4)
D. f(x)=|x|
Answer
C. √(x−4) is a transformation, not the pure parent.
18. If f(x)=|x|, then f(−3)=?
A. −3
B. 0
C. 3
D. undefined
Answer
C. Absolute value outputs 3.
19. The domain of f(x)=1/x is:
A. All reals
B. All reals except 0
C. x > 0
D. x ≥ 0
Answer
B. Reciprocal undefined at 0.
20. Which function has a horizontal asymptote?
A. f(x)=x²
B. f(x)=1/x
C. f(x)=|x|
D. f(x)=√x
Answer
B. Reciprocal approaches 0 as |x| grows.
21. Which of these is odd?
A. f(x)=x³
B. f(x)=x²
C. f(x)=|x|
D. f(x)=√x
Answer
A. Cubic is symmetric about origin.
22. What is the vertex of f(x)=(x−2)²+3?
A. (0,0)
B. (2,3)
C. (−2,3)
D. (2,−3)
Answer
B. Vertex at (h,k) = (2,3).
23. Which is the inverse of f(x)=√x?
A. f⁻¹(x)=x²
B. f⁻¹(x)=1/x²
C. f⁻¹(x)=√x
D. f⁻¹(x)=x³
Answer
A. Swapping gives y²=x → x².
24. Which parent function is always increasing?
A. f(x)=√x
B. f(x)=x³
C. f(x)=x²
D. f(x)=|x|
Answer
B. Cubic increases over all real numbers.
25. Which function has range (−∞,∞)?
A. f(x)=√x
B. f(x)=x²
C. f(x)=x³
D. f(x)=|x|
Answer
C. Only cubic covers all reals.
26. If f(x)=√(x−1), what is the correct domain?
A. x ≥ 0
B. x ≥ 1
C. x > 1
D. x ≤ 1
Answer
B. Inside sqrt requires x−1 ≥ 0 → x ≥ 1.
27. The inverse of f(x)=(x−4)³+2 is:
A. f⁻¹(x)=(x−2)^(1/3)+4
B. f⁻¹(x)=(x+2)^(1/3)+4
C. f⁻¹(x)=(x−2)^(1/3)−4
D. f⁻¹(x)=(x+2)^(1/3)−4
Answer
A. Swap x,y: x=(y−4)³+2 → (x−2)^(1/3)+4.
28. Which sequence of transformations produces g(x)=−2(x+1)²+5 from f(x)=x²?
A. Left 1, reflect x-axis, vertical stretch by 2, up 5
B. Right 1, reflect x-axis, vertical stretch by 2, up 5
C. Left 1, vertical shrink by 1/2, up 5
D. Right 1, vertical stretch by 2, down 5
Answer
A. (x+1) shift left, “−” reflection, 2 multiplier stretch, +5 upward.
29. Which parent function is one-to-one without restriction?
A. f(x)=x²
B. f(x)=√x
C. f(x)=x³
D. f(x)=|x|
Answer
C. Cubic passes horizontal line test globally.
30. Which graph represents the reciprocal of f(x)=x?
A. Line through origin
B. Hyperbola in quadrants I & III
C. Parabola opening upward
D. Absolute value V-shape
Answer
B. Reciprocal yields 1/x, hyperbola.
31. Which condition ensures two functions are inverses?
A. f(g(x))=f(x)
B. g(f(x))=f(x)
C. f(g(x))=g(f(x))=x
D. f(g(x))=x²
Answer
C. Inverses must compose to identity x.
32. For h(t)=−16t²+64t+80, the maximum height is:
A. 80
B. 144
C. 160
D. 208
Answer
C. Vertex at t=−b/(2a)=2; h(2)=160.
33. Which describes the range of f(x)=1/(x−3)?
A. All real numbers
B. All real numbers except 3
C. All real numbers except 0
D. y ≠ 0
Answer
D. Reciprocal outputs never equal 0.
34. Which of the following functions is odd?
A. f(x)=x²+1
B. f(x)=−x³
C. f(x)=|x|
D. f(x)=√x
Answer
B. Negative cubic retains origin symmetry.
35. The inverse of f(x)=(x+2)², x ≥ −2 is:
A. f⁻¹(x)=√x−2
B. f⁻¹(x)=√(x−2)
C. f⁻¹(x)=√x+2
D. f⁻¹(x)=−√x−2
Answer
C. Swap x,y: x=(y+2)² → y=√x−2; with domain restriction x≥−2, solution is √x−2.
36. Find the domain of f(x)=√(2x−6).
Answer
Require 2x−6 ≥ 0 → x ≥ 3. Domain is [3,∞).
37. Determine the inverse of f(x)=3x−7.
Answer
Swap x,y → x=3y−7 → y=(x+7)/3. So f⁻¹(x)=(x+7)/3.
38. A ball is thrown upward with h(t)=−16t²+96t+64. What is the maximum height?
Answer
Vertex at t=−b/(2a)=96/(32)=3. h(3)=−16(9)+288+64=208. Max height=208.
39. Solve for x: f(x)=|x−4|=7.
Answer
x−4=±7 → x=11 or x=−3.
40. State the range of f(x)=−x²+9.
Answer
Parabola opens down, vertex (0,9). Range: (−∞,9].
41. If f(x)=x³, evaluate f⁻¹(27).
Answer
Inverse is cube root. f⁻¹(27)=³√27=3.
42. Find the x-intercepts of f(x)=x²−5x+6.
Answer
Factor (x−2)(x−3)=0 → intercepts x=2,3.
43. Solve: √(x+1)=5.
Answer
x+1=25 → x=24.
44. Determine if f(x)=x² is one-to-one.
Answer
No. f(2)=4 and f(−2)=4; fails horizontal line test.
45. Find the vertex of f(x)=(x+1)²−4.
Answer
Vertex (−1,−4).
46. Write the equation of the line of symmetry for f(x)=x²−6x+5.
Answer
x=−b/(2a)=6/(2)=3. Symmetry axis: x=3.
47. Find the domain of f(x)=1/(x²−9).
Answer
Denominator=0 at x=±3. Domain: (−∞,−3)∪(−3,3)∪(3,∞).
48. If g(x)=2x+1, find g⁻¹(9).
Answer
Inverse: (x−1)/2. g⁻¹(9)=(9−1)/2=4.
49. State the end behavior of f(x)=−2x³+5.
Answer
As x→∞, f(x)→−∞; as x→−∞, f(x)→∞.
50. Solve for x: log₂(x)=5.
Answer
x=2⁵=32.