Rucete ✏ AP Precalculus In a Nutshell
4. Parent Functions — Practice Questions
This chapter explores parent functions, their transformations, compositions, and inverses.
(Multiple Choice — Click to Reveal Answer)
1. Which of the following is the graph of the parent function f(x) = x^2?
(A) A straight line through the origin
(B) A U-shaped parabola opening upward
(C) A V-shaped graph opening upward
(D) A horizontal line
Answer
(B) — The parent quadratic function is a parabola opening upward with vertex at (0,0).
2. The graph of f(x) = |x| is best described as:
(A) U-shaped
(B) V-shaped
(C) Linear
(D) Cubic
Answer
(B) — The absolute value function forms a V shape with vertex at the origin.
3. What is the domain of f(x) = √x?
(A) All real numbers
(B) x ≥ 0
(C) x ≤ 0
(D) x > 0
Answer
(B) — The square root function is defined only for nonnegative x-values.
4. Which parent function has range y ≥ 0?
(A) f(x) = x
(B) f(x) = x^3
(C) f(x) = |x|
(D) f(x) = 1/x
Answer
(C) — The absolute value function outputs only nonnegative values.
5. Which of the following represents the reciprocal function?
(A) f(x) = x
(B) f(x) = 1/x
(C) f(x) = √x
(D) f(x) = x^2
Answer
(B) — The reciprocal parent function is f(x) = 1/x.
6. The parent cubic function f(x) = x^3 is symmetric about:
(A) The x-axis
(B) The y-axis
(C) The origin
(D) None
Answer
(C) — The cubic function is odd, so it has origin symmetry.
7. Which parent function has a horizontal asymptote?
(A) f(x) = x^2
(B) f(x) = |x|
(C) f(x) = 1/x
(D) f(x) = x^3
Answer
(C) — f(x) = 1/x has a horizontal asymptote at y=0.
8. The graph of f(x) = √x starts at:
(A) (0,0)
(B) (1,1)
(C) (−1,0)
(D) (0,1)
Answer
(A) — At x=0, f(0)=0.
9. Which parent function passes both the vertical line test and the horizontal line test?
(A) f(x) = x^2
(B) f(x) = |x|
(C) f(x) = x^3
(D) f(x) = √x
Answer
(C) — f(x) = x^3 is one-to-one, passing both tests.
10. The vertex of the parent quadratic function f(x) = x^2 is:
(A) (0,0)
(B) (1,0)
(C) (0,1)
(D) (1,1)
Answer
(A) — The parabola opens upward with vertex at the origin.
11. Which function is even?
(A) f(x) = x^3
(B) f(x) = x^2
(C) f(x) = √x
(D) f(x) = 1/x
Answer
(B) — Quadratic is symmetric about the y-axis, so it is even.
12. Which function is odd?
(A) f(x) = x^2
(B) f(x) = |x|
(C) f(x) = x^3
(D) f(x) = √x
Answer
(C) — Cubic functions satisfy f(−x) = −f(x).
13. The reciprocal function f(x) = 1/x is undefined at:
(A) x = 1
(B) x = 0
(C) x = −1
(D) x = 2
Answer
(B) — Division by zero is undefined.
14. Which parent function has the range y > 0 only?
(A) f(x) = x^2
(B) f(x) = √x
(C) f(x) = |x|
(D) f(x) = e^x
Answer
(D) — The exponential function is always positive.
15. Which parent function’s graph includes the point (1,1), (2,4), (3,9)?
(A) f(x) = x
(B) f(x) = x^2
(C) f(x) = √x
(D) f(x) = 1/x
Answer
(B) — Squaring the inputs matches the outputs.
16. What is the y-intercept of f(x) = 1/x?
(A) None
(B) 1
(C) 0
(D) −1
Answer
(A) — Reciprocal function has no value at x=0, so no y-intercept.
17. Which parent function has domain (−∞, ∞) and range (−∞, ∞)?
(A) f(x) = √x
(B) f(x) = |x|
(C) f(x) = x
(D) f(x) = 1/x
Answer
(C) — The identity function covers all real numbers.
18. Which is the inverse of f(x) = √x, x ≥ 0?
(A) f−1(x) = x^2
(B) f−1(x) = √x
(C) f−1(x) = 1/x
(D) f−1(x) = |x|
Answer
(A) — Swapping variables and squaring gives the inverse on restricted domain.
19. The graph of f(x) = |x| has its minimum at:
(A) (1,0)
(B) (0,0)
(C) (−1,0)
(D) (0,1)
Answer
(B) — Vertex of absolute value graph is at origin.
20. Which of the following is a linear parent function?
(A) f(x) = x^2
(B) f(x) = |x|
(C) f(x) = x
(D) f(x) = √x
Answer
(C) — Identity function f(x)=x is linear.
21. Which parent function decreases on (−∞,0) and increases on (0,∞)?
(A) f(x) = |x|
(B) f(x) = x^3
(C) f(x) = √x
(D) f(x) = x^2
Answer
(A) — Absolute value decreases until the origin, then increases.
22. The reciprocal function f(x) = 1/x has how many branches?
(A) 1
(B) 2
(C) 3
(D) 4
Answer
(B) — It has one branch in quadrant I and another in quadrant III.
23. Which parent function is not continuous?
(A) f(x) = x
(B) f(x) = |x|
(C) f(x) = 1/x
(D) f(x) = x^2
Answer
(C) — Reciprocal function has a discontinuity at x=0.
24. Which of the following is the range of f(x) = √x?
(A) (−∞, ∞)
(B) [0, ∞)
(C) (−∞, 0]
(D) (0, ∞)
Answer
(B) — Square root outputs only nonnegative values.
25. Which function has a maximum turning point at the origin?
(A) f(x) = x^2
(B) f(x) = |x|
(C) f(x) = −x^2
(D) f(x) = √x
Answer
(C) — f(x)=−x^2 opens downward, vertex at (0,0) is a maximum.
26. Which transformation produces g(x) = −f(x) if f(x) = x^2?
(A) Reflection across the y-axis
(B) Reflection across the x-axis
(C) Shift upward
(D) Shift downward
Answer
(B) — Negating the function reflects it across the x-axis.
27. If h(x) = (x−2)^2 + 3, which describes the transformation from f(x)=x^2?
(A) Left 2, up 3
(B) Right 2, up 3
(C) Left 2, down 3
(D) Right 2, down 3
Answer
(B) — (x−2)^2 shifts right 2, +3 shifts up 3.
28. What is the inverse of f(x) = 3x+5?
(A) f−1(x) = (x−5)/3
(B) f−1(x) = 3x−5
(C) f−1(x) = x/3+5
(D) f−1(x) = (x+5)/3
Answer
(A) — Swap x,y and solve: y = (x−5)/3.
29. Which is true about the graph of f(x)=√(x−4)?
(A) Starts at (0,0)
(B) Starts at (4,0)
(C) Starts at (−4,0)
(D) Starts at (0,4)
Answer
(B) — Inside shift moves starting point to (4,0).
30. Which composite satisfies f(g(x)) = g(f(x))?
(A) f(x)=2x, g(x)=x+1
(B) f(x)=x^2, g(x)=√x
(C) f(x)=3x+k, g(x)=1/x
(D) None
Answer
(C) — Only with certain k values can compositions be equal.
31. Which function has a slant (oblique) asymptote?
(A) f(x) = (x^2+1)/(x+1)
(B) f(x) = x^2
(C) f(x) = √x
(D) f(x) = 1/x
Answer
(A) — Degree numerator (2) > denominator (1) → slant asymptote.
32. Which transformation produces f(x)=|x−3|?
(A) Shift left 3
(B) Shift right 3
(C) Shift up 3
(D) Shift down 3
Answer
(B) — |x−3| shifts absolute value graph right 3.
33. Which of the following represents exponential growth?
(A) f(x) = (1/2)^x
(B) f(x) = 2^x
(C) f(x) = −x^2
(D) f(x) = |x|
Answer
(B) — Base > 1 indicates growth.
34. Which parent function is used to model inverse variation?
(A) f(x) = √x
(B) f(x) = x^2
(C) f(x) = 1/x
(D) f(x) = x^3
Answer
(C) — Reciprocal function models inverse variation.
35. Which statement about inverse functions is true?
(A) Their graphs are reflections across the x-axis
(B) Their graphs are reflections across the y=x line
(C) Their graphs are translations
(D) They have no relation
Answer
(B) — Inverse functions reflect across the line y=x.
36. Find the domain of f(x) = 1/(x−4).
Answer
All real numbers except x=4. The denominator cannot be zero.
37. Determine the range of f(x) = √(x+2).
Answer
[0, ∞). Square roots output only nonnegative values.
38. Find the inverse of f(x) = 2x−7.
Answer
f−1(x) = (x+7)/2. Swap x and y, then solve for y.
39. Identify the vertex of f(x) = (x+3)^2 − 5.
Answer
(−3, −5). Shifts left 3, down 5 from the parent parabola.
40. Solve f(x) = |x| for f(x) = 4.
Answer
x=4 or x=−4. Absolute value equals 4 at ±4.
41. State the asymptotes of f(x) = 1/(x+2).
Answer
Vertical asymptote: x=−2; horizontal asymptote: y=0.
42. Find the y-intercept of f(x) = 3x^2 − 2x + 1.
Answer
(0,1). Substitute x=0.
43. If f(x) = x^3, compute f(−2).
Answer
−8. Cube of −2 is −8.
44. Solve for x: √x = 7.
Answer
x=49. Square both sides.
45. Find the slope of the linear parent function f(x) = x.
Answer
Slope = 1.
46. Determine the x-intercepts of f(x) = x^2 − 9.
Answer
(−3,0) and (3,0). Factor to (x−3)(x+3).
47. If g(x) = 2^x, evaluate g(3).
Answer
8. Because 2^3 = 8.
48. State whether f(x)=x^3 is even, odd, or neither.
Answer
Odd. f(−x) = −f(x).
49. Find f−1(x) if f(x)=√(x−1), x ≥ 1.
Answer
f−1(x)=x^2+1. Swap x,y and solve.
50. A ball’s height is modeled by h(t) = −16t^2 + 64t + 80. What is the maximum height?
Answer
Maximum at t=2 seconds. h(2)=144 feet.