Rucete ✏ AP Precalculus In a Nutshell
3. Rational Functions — Practice Questions
This chapter introduces key features of rational functions including domain, intercepts, holes, asymptotes, end behavior, and rational inequalities.
(Multiple Choice — Click to Reveal Answer)
1. Which defines a rational function?
(A) Sum of two polynomials
(B) Quotient of two polynomials with polynomial denominator not identically 0
(C) Any differentiable function
(D) Any function with asymptotes
Answer
(B) — By definition, a rational function is a ratio of two polynomials with a nonzero polynomial denominator.
2. The domain of f(x) = (x + 2) / (x - 3) is
(A) all real numbers
(B) all real numbers except 3
(C) all real numbers except -2
(D) all real numbers except -2 and 3
Answer
(B) — Denominator 0 at x = 3 is excluded.
3. Vertical asymptote of f(x) = 1 / (x + 5):
(A) x = -5
(B) y = -5
(C) x = 5
(D) y = 5
Answer
(A) — Denominator 0 at x = -5.
4. Horizontal asymptote of f(x) = (3x^2 + 1) / (x^2 + 7):
(A) y = 0
(B) y = 1
(C) y = 3
(D) none
Answer
(C) — Degrees equal; ratio of leading coefficients 3/1.
5. For f(x) = (x^2 - 9) / (x - 3), the behavior at x = 3 is
(A) vertical asymptote
(B) hole (removable)
(C) x-intercept
(D) continuous
Answer
(B) — Factor to (x - 3)(x + 3)/(x - 3); cancellation yields a hole at x = 3.
6. Which has horizontal asymptote y = 0?
(A) (x^2 + 1)/(x^3 + 2)
(B) (3x^3 + 1)/(2x^3 + 7)
(C) (2x^3)/(x^2 + 1)
(D) (x^2 + x)/(x^2 - 5)
Answer
(A) — Degree numerator (2) < denominator (3).
7. Slant asymptote of f(x) = (x^2 + 1)/(x - 1) is
(A) y = x + 1
(B) y = x
(C) y = x - 1
(D) y = 0
Answer
(B) — Division gives quotient x with remainder; slant asymptote y = x.
8. y-intercept of f(x) = (2x - 4)/(x + 1):
(A) -2
(B) -4
(C) 0
(D) 2
Answer
(B) — f(0) = -4/(1) = -4.
9. End behavior asymptote of f(x) = (x^2 - 1)/(x^2 + 1):
(A) y = 0
(B) y = 1
(C) y = x
(D) none
Answer
(B) — Degrees equal → ratio 1.
10. Vertical asymptotes of g(x) = (x^2 - 4)/(x^2 - 5x + 6):
(A) x = 2, 3
(B) x = -2, 3
(C) x = 1, 2
(D) x = -3, -2
Answer
(A) — Denominator factors (x - 2)(x - 3).
11. Which has a hole at x = 2?
(A) (x^2 - 4)/(x - 2)
(B) (x^2 - 4)/(x^2 - 2)
(C) (x^2 - 4)/(x^2 - 4)
(D) (x^2 - 4)/(x + 2)
Answer
(A) — Common factor (x - 2) cancels with denominator.
12. Which statement is true?
(A) A graph never crosses a horizontal asymptote
(B) A graph never crosses a slant asymptote
(C) A graph may cross horizontal or slant asymptotes
(D) A graph may cross vertical asymptotes
Answer
(C) — Vertical asymptotes cannot be crossed; others can.
13. x-intercept of f(x) = (2x + 4)/(x^2 + 1):
(A) -2
(B) 2
(C) 4
(D) none
Answer
(A) — Set numerator 0 → x = -2.
14. Solution of (x - 1)/(x + 2) > 0:
(A) (-∞, -2) ∪ (1, ∞)
(B) (-2, 1)
(C) (1, ∞)
(D) (-∞, -2)
Answer
(A) — Critical points -2 (VA), 1 (zero); sign chart yields two positive intervals.
15. End behavior of f(x) = (5x^3 + 1)/(2x^3 - 7):
(A) approaches y = 5/2
(B) approaches y = 0
(C) unbounded
(D) slant asymptote
Answer
(A) — Equal degrees → ratio 5/2.
16. For f(x) = (x^2 - 1)/(x - 1), x = 1 is
(A) vertical asymptote
(B) removable discontinuity (hole)
(C) zero of f
(D) in the domain
Answer
(B) — Cancellation leaves a hole.
17. Which has no horizontal asymptote?
(A) (x^3 + 1)/(x^2 + 2)
(B) (2x^2 + 1)/(x^2 + 3)
(C) (x + 1)/(x^2 + 1)
(D) (x^2 + 2)/(2x^2 + 7)
Answer
(A) — Degree numerator > denominator → slant (or higher) asymptote, not horizontal.
18. Solve (x + 1)/(x - 3) = 0.
(A) x = -1
(B) x = 3
(C) no solution
(D) x = 0
Answer
(A) — Zero occurs when numerator 0 and denominator ≠ 0.
19. Vertical asymptotes of f(x) = 4/(x^2 - 9):
(A) x = ±3
(B) x = 0
(C) y = ±3
(D) y = 0
Answer
(A) — Denominator 0 at x = ±3.
20. Which is always possible for rational functions?
(A) Be defined at all real numbers
(B) Have vertical, horizontal, or slant asymptotes
(C) Have no intercepts
(D) Be linear only
Answer
(B) — Asymptotes are common features of rational functions.
21. y-intercept of f(x) = (x^2 + 2)/(x^2 + 4):
(A) 0.5
(B) 1
(C) 2
(D) none
Answer
(A) — f(0) = 2/4 = 1/2.
22. For f(x) = (x - 2)/x, which is true on (0, 2)?
(A) f(x) > 0
(B) f(x) < 0
(C) f(x) = 0
(D) undefined
Answer
(B) — Numerator negative, denominator positive.
23. End behavior of f(x) = (x^3 - 2)/(x^2 + 5):
(A) y = 0
(B) unbounded both ends
(C) slant asymptote (linear)
(D) horizontal asymptote y = 1
Answer
(C) — Degree difference 1 → linear slant asymptote.
24. As x → ∞, (x^2 - 4)/(x^2 + 4) approaches
(A) 0
(B) 1
(C) 2
(D) ∞
Answer
(B) — Ratio 1.
25. For f(x) = 1/(x^2 + 1), which is true?
(A) Has vertical asymptotes
(B) Has no real zeros
(C) Has a hole
(D) Has a slant asymptote
Answer
(B) — Numerator never 0, denominator never 0 for real x.
26. Let f(x) = (x^2 - 6x + 8)/(x^2 - 5x + 6). Identify discontinuities.
(A) VAs at x = 2, 3; no holes
(B) Hole at x = 2; VA at x = 3
(C) VA at x = 2; hole at x = 3
(D) Holes at x = 2 and 3
Answer
(B) — Factor: (x-2)(x-4)/[(x-2)(x-3)] → hole at x = 2, VA at x = 3.
27. A rational function has HA y = 2 and VA x = -4. Which could be its formula?
(A) (2x - 3)/(x + 4)
(B) (x + 4)/(2x - 3)
(C) (2x^2 + 1)/(x^2 + 4x + 16)
(D) (2x^2 + 5)/(x^2 + 8x + 16)
Answer
(A) — Linear/linear with ratio 2 and denominator zero at x = -4 satisfies both.
28. For h(x) = (3x^3 - x + 1)/(x^2 - 1), the end-behavior asymptote is
(A) y = 3x + 3
(B) y = 3x - 3
(C) y = 3x
(D) y = x + 3
Answer
(C) — Degree difference 1; division gives linear quotient with leading term 3x.
29. Suppose f(x) = (x^2 - 1)/(x^2 - 4). Which is true?
(A) VAs at x = ±1
(B) Holes at x = ±1
(C) VAs at x = ±2
(D) Hole at x = 2, VA at x = -2
Answer
(C) — Denominator zeros at x = ±2; no common factor with numerator.
30. Which function has a hole at x = -1 and VA at x = 4?
(A) (x + 1)(x - 2)/(x - 4)
(B) (x + 1)/(x + 1)
(C) (x + 1)(x - 3)/[(x + 1)(x - 4)]
(D) (x - 3)/(x - 4)
Answer
(C) — Common factor (x + 1) cancels (hole at -1); remaining denominator (x - 4) → VA at 4.
31. Let r(x) = (2x^2 + 5x - 3)/(x - 1). The slant asymptote is
(A) y = 2x + 7
(B) y = 2x + 9
(C) y = 2x + 2
(D) y = 2x + 5
Answer
(C) — Divide: (2x^2 + 5x - 3) ÷ (x - 1) = 2x + 7 with remainder 4 → slant y = 2x + 7. Correction: compute carefully: 2x^2/(x)=2x; multiply back 2x(x-1)=2x^2-2x; subtract → 7x-3; next +7; 7(x-1)=7x-7; remainder 4. So slant is y = 2x + 7. The correct choice is (A).
32. For p(x) = (x^3 - 4x)/(x^2 - 1), which is true about intercepts?
(A) x-intercepts at x = -2, 0, 2; y-intercept 0
(B) x-intercepts at x = -1, 0, 1; y-intercept 0
(C) x-intercepts at x = -2, 0, 2; y-intercept -4
(D) x-intercepts at x = -1, 0, 1; y-intercept 1
Answer
(A) — Numerator x(x-2)(x+2)=0 gives x = -2, 0, 2; p(0)=0.
33. Solve the inequality (x + 1)/(x - 2) ≤ 0.
(A) (-∞, -1] ∪ (2, ∞)
(B) [-1, 2)
(C) (-∞, -1] ∪ (2, ∞)
(D) (-∞, -1] ∪ (2, ∞) with 2 included
Answer
(B) — Critical points -1 (zero, included) and 2 (VA, excluded); sign chart gives [-1, 2).
34. A rational function crosses its horizontal asymptote. Which is possible?
(A) (x^2 + 1)/(x^2 - 1)
(B) (x^2 - 4)/(x^2 + 1)
(C) (x + 1)/x
(D) 1/x
Answer
(B) — Equal degrees with HA y = 1; mid-graph can cross HA.
35. For q(x) = (x^2 + 3x + 2)/(x^2 + x - 2), which is true?
(A) Hole at x = -1; VA at x = 2
(B) VA at x = -1; hole at x = 2
(C) VAs at x = -1, 2; no holes
(D) Hole at x = -1; HA y = 0
Answer
(C) — Denominator factors (x + 2)(x - 1); numerator (x + 1)(x + 2). No common factor with (x - 1), so VAs at x = -2 and 1. Note: options use -1,2; correct statement among options is none. To align, interpret q(x) = (x^2 + 3x + 2)/(x^2 + x - 2) = (x+1)(x+2)/[(x+2)(x-1)] → there is a common (x+2). So hole at x = -2, VA at x = 1. Replace options: Correct choice is “Hole at x = -2; VA at x = 1.”
36. Find all vertical asymptotes of f(x) = (x^2 + 1)/(x^2 - 9).
Answer
x = -3, 3 — Denominator zero where x^2 - 9 = 0.
37. Determine the hole (if any) of g(x) = (x^2 - 1)/(x^2 - x - 2).
Answer
Hole at x = 1; y = limit value 2 — Factor: (x-1)(x+1)/[(x-2)(x-1)] → cancel; plug x=1 into reduced (x+1)/(x-2) to get 2/(-1) = -2. Correction: (x+1)/(1-2) = - (x+1) at x=1 gives -2. So hole at (1, -2).
38. Find the horizontal asymptote of h(x) = (4x^3 - x)/(2x^3 + 5).
Answer
y = 2 — Equal degrees, ratio 4/2 = 2.
39. Long divide to get the slant asymptote of r(x) = (x^2 - 5x + 7)/(x - 2).
Answer
y = x - 3 — Division gives quotient x - 3 with remainder 1.
40. Solve (x + 3)/(x - 1) < 0.
Answer
(-3, 1) — Zero at -3 (excluded), VA at 1 (excluded); sign chart shows negative between them.
41. Find x-intercepts of f(x) = (x^2 - 4x)/(x^2 + 1).
Answer
x = 0 and x = 4 — Numerator x(x - 4) = 0.
42. Determine y-intercept of p(x) = (3x - 2)/(x^2 - 4).
Answer
y = 1/2 — p(0) = (-2)/(-4) = 1/2.
43. For s(x) = (x^2 + 1)/(x + 1), state the end behavior asymptote.
Answer
y = x - 1 — Degree difference 1; division yields linear asymptote.
44. Give intervals where f(x) = (x - 2)/(x + 1) > 0.
Answer
(-∞, -1) ∪ (2, ∞) — Signs same when both numerator and denominator have same sign.
45. For t(x) = (x^2 - 1)/(x^2 - 2x - 3), list VAs and holes.
Answer
VA at x = 3; hole at x = -1 with y = limit value. Factor: numerator (x-1)(x+1); denominator (x-3)(x+1). Cancel (x+1) → hole at x=-1; reduced (x-1)/(x-3) gives y = (-2)/(-4) = 1/2 at the hole.
46. Compute lim (x→∞) (2x^2 - 5x + 1)/(x^2 + x + 9).
Answer
2 — Ratio of leading coefficients 2/1.
47. Compute lim (x→-∞) (x^3 - 4x)/(x^2 + 1).
Answer
-∞ — Degree numerator greater; leading behavior like x.
48. Solve (x^2 - 9)/(x - 1) ≥ 0.
Answer
(-∞, -3] ∪ [3, ∞) ∪ (1, 3) where sign is nonnegative; proceed via sign chart on critical points x = -3, 1, 3, with x = 1 excluded.
49. Find all intercepts of f(x) = (x - 3)/(x^2 - 9).
Answer
x-intercept x = 3 is a hole, not an intercept; y-intercept at x = 0 gives y = -3/(-9) = 1/3. Factor shows common (x - 3) cancels → hole at x=3; VA at x = -3.
50. Sketching guide for g(x) = (x^2 - 16)/(x^2 - 4): list HAs, VAs, holes.
Answer
HA: y = 1 (equal degrees). Factor: numerator (x - 4)(x + 4); denominator (x - 2)(x + 2). No common factor → no holes. VAs at x = -2, 2.