Rucete ✏ AP Precalculus In a Nutshell
3. Rational Functions — Practice Questions 3
This chapter explores rational functions with new question types: modeling, symmetry, limits, discontinuities, and inequalities.
(Multiple Choice — Click to Reveal Answer)
1. Which function’s domain excludes x = -3?
(A) (x^2+1)/(x+3)
(B) (x-3)/(x+2)
(C) (x^2-9)/(x^2+1)
(D) (x+1)/(x^2+9)
Answer
(A) — Denominator zero at x = -3, so excluded.
2. Horizontal asymptote of f(x) = (4x^2+3)/(2x^2-5) is
(A) y=0
(B) y=2
(C) y=1/2
(D) y=4
Answer
(B) — Equal degree; leading-coefficient ratio 4/2 = 2.
3. Which function has an oblique (slant) asymptote?
(A) (x^3+1)/(x^2+1)
(B) (x^2+1)/(x^3+1)
(C) (2x^2+1)/(2x^2+3)
(D) (x+1)/(x^2+1)
Answer
(A) — Degree(numerator) − Degree(denominator) = 1 → slant asymptote.
4. Which function has a hole at x = 5?
(A) (x-5)(x+2)/(x-5)(x-3)
(B) (x^2-25)/(x-5)
(C) (x+5)/(x^2-25)
(D) (x-1)/(x+5)
Answer
(A) — Factor (x−5) cancels → removable discontinuity (hole) at x=5.
5. For g(x) = (x^2-9)/(x^2-4), vertical asymptotes are
(A) x = ±3
(B) x = ±2
(C) x = 2 only
(D) none
Answer
(B) — Denominator 0 at x=±2 and no cancellation with numerator.
6. y-intercept of f(x) = (x^2+4x+4)/(x^2+1) is
(A) 0
(B) 1
(C) 2
(D) 4
Answer
(D) — f(0) = 4/1 = 4.
7. Which function approaches y=0 as x→∞?
(A) (2x+1)/(x^3+4)
(B) (x^2+1)/(2x^2+1)
(C) (3x^3+1)/(x^2+1)
(D) (x^2-1)/(x^2-1)
Answer
(A) — Degree(numerator) < Degree(denominator) → HA y=0.
8. Which graph is symmetric about the origin (odd)?
(A) x/(x^2+1)
(B) (x^2+1)/(x^2+2)
(C) (x-1)/(x+1)
(D) (x^4+2)/(x^2+1)
Answer
(A) — f(-x)=−f(x) → odd symmetry.
9. Which function has no real zeros?
(A) (x^2+1)/(x^2+4)
(B) (x-2)/(x+3)
(C) (x^2-9)/(x^2+1)
(D) (x+1)/(x^2-1)
Answer
(A) — Numerator x^2+1 ≠ 0 for real x.
10. lim (x→∞) (x^2-1)/(x^2+1) equals
(A) 0
(B) 1
(C) -1
(D) ∞
Answer
(B) — Ratio of leading coefficients = 1.
11. Which has a removable discontinuity?
(A) (x-2)(x+3)/(x-2)(x+5)
(B) (x+2)/(x-2)
(C) (x^2+1)/(x+1)
(D) (x^3+1)/(x^2+1)
Answer
(A) — (x−2) cancels → hole at x=2.
12. Which function has a vertical asymptote at x = −2?
(A) (x+2)/(x-3)
(B) (x^2+1)/(x+2)
(C) (x^2-4)/(x^2+9)
(D) (x-1)/(x+1)
Answer
(B) — Denominator zero at x = −2.
13. lim (x→−∞) (5x^3+2)/(x^3-1) equals
(A) 5
(B) −5
(C) 0
(D) 1
Answer
(A) — Leading-coefficient ratio 5/1.
14. Which rational function is continuous for all real x?
(A) (x^2+1)/(x^2+4)
(B) (x-3)/(x^2-9)
(C) (x+1)/(x-1)
(D) 1/x
Answer
(A) — Denominator x^2+4 ≠ 0 for all real x.
15. Which has slant asymptote y = 2x + 1?
(A) (2x^2+x+1)/(x+1)
(B) (x^2+1)/(x+2)
(C) (3x^2-1)/(x+3)
(D) (x^3+1)/(x^2+1)
Answer
(A) — Long division yields quotient 2x+1.
16. For f(x)=(x^2+9)/(x^2−16), the horizontal asymptote is
(A) y=0
(B) y=1
(C) y=9/16
(D) y=−1
Answer
(B) — Equal degree → ratio 1.
17. Which is symmetric about the y-axis (even)?
(A) (x^2+1)/(x^4+3)
(B) (x+1)/(x-1)
(C) (x^3+1)/(x^2+1)
(D) (x-2)/(x+2)
Answer
(A) — Only even powers of x → f(−x)=f(x).
18. x-intercepts of (x^2−4)/(x^2+1) are
(A) x=±2
(B) x=0
(C) none
(D) x=±1
Answer
(A) — Numerator zero at x=±2.
19. For f(x)=1/(x^2+1), which is true?
(A) Has a hole
(B) Domain excludes 0
(C) Range is (0,1]
(D) Horizontal asymptote y=0
Answer
(D) — Degree(numerator) < Degree(denominator) → HA y=0.
20. Which has vertical asymptotes but no holes?
(A) (x^2−9)/(x^2−4)
(B) (x−2)(x+2)/(x^2−4)
(C) (x+1)/(x−1)
(D) (x^2+1)/(x^2+9)
Answer
(A) — Denominator zeros at ±2; no cancellation with numerator.
21. lim (x→3) (x^2−9)/(x−3) equals
(A) 6
(B) ∞
(C) 0
(D) −6
Answer
(A) — Simplifies to x+3; at x=3 gives 6 (hole at x=3 in original).
22. Which has numerator degree smaller than denominator?
(A) (x^2+1)/(x^3+1)
(B) (x^3+1)/(x^2+1)
(C) (2x^2+1)/(x^2+1)
(D) (x^2−1)/(x^2+1)
Answer
(A) — 2 vs 3.
23. Which has no vertical asymptotes?
(A) (x^2+1)/(x^2+4)
(B) (x+1)/(x−1)
(C) (x^2−9)/(x^2−4)
(D) 1/x
Answer
(A) — Denominator never zero.
24. End behavior of (5x^4+2)/(2x^4−7) is
(A) y=0
(B) y=5/2
(C) y=2/5
(D) y=∞
Answer
(B) — Ratio of leading coefficients 5/2.
25. For f(x)=(x^2−25)/(x^2+5x+6), discontinuities are
(A) hole at x=−2; VA at x=−3
(B) hole at x=5; VA at x=3
(C) VAs at x=−2, −3
(D) none
Answer
(C) — Denominator factors (x+2)(x+3) with no cancellation.
26. Which has both a hole and a vertical asymptote?
(A) (x−2)(x+3)/(x−2)(x−4)
(B) (x^2+1)/(x+5)
(C) (x−1)/(x^2+1)
(D) (x^2−4)/(x^2+9)
Answer
(A) — (x−2) cancels → hole at x=2; remaining (x−4) → VA at x=4.
27. Slant asymptote of f(x)=(2x^3+5x^2−1)/(x^2+1) is
(A) y=2x+5
(B) y=2x+3
(C) y=2x+1
(D) y=x+2
Answer
(A) — Long division quotient 2x+5.
28. As x→∞, (x^2−4)/(x^2−9) approaches
(A) 0
(B) 1
(C) 2
(D) no HA
Answer
(B) — Equal degree; ratio 1.
29. Which has oblique asymptote y=3x−2?
(A) (3x^2−2x+1)/(x+1)
(B) (3x^2−2)/(x+2)
(C) (3x^2+1)/(x−1)
(D) (3x^2−6x+2)/(x+1)
Answer
(A) — Division yields quotient 3x−2.
30. For h(x)=(x^3−2x)/(x^2−1), vertical asymptotes are
(A) x=±1
(B) x=0,1
(C) x=2
(D) none
Answer
(A) — Denominator (x−1)(x+1)=0 at ±1.
31. Which graph can intersect its slant asymptote?
(A) (x^2+1)/(x+1)
(B) (3x^3+1)/(x^2+2)
(C) (2x^2+1)/(x+2)
(D) (x^3+2)/(x^2+1)
Answer
(D) — Rational graphs may cross horizontal/slant asymptotes.
32. Inequality (x+1)/(x−2) > 0 has solution set
(A) (−∞,−1) ∪ (2,∞)
(B) (−∞,−1) ∪ (2,∞) including 2
(C) (−1,2)
(D) (2,∞) only
Answer
(A) — Zeros at −1 (included if ≥), VA at 2 (excluded); sign chart → (−∞,−1)∪(2,∞).
33. lim (x→2⁺) (x^2−4)/(x−2) equals
(A) −∞
(B) +∞
(C) 0
(D) does not exist
Answer
(B) — (x−2)(x+2)/(x−2)=x+2 but denominator→0⁺, so → +∞.
34. For f(x)=(x^2+1)/(x−3), the slant asymptote is
(A) y=0
(B) y=x+3
(C) y=x+3 with remainder
(D) y=x−3
Answer
(B) — Long division quotient x+3; that line is the slant asymptote.
35. Symmetry of f(x)=(x^2−1)/(x^4+1) is
(A) even
(B) odd
(C) neither
(D) both
Answer
(A) — All even powers → f(−x)=f(x).
36. Find all vertical asymptotes of f(x)=(x^2+1)/(x^2−16).
Answer
x = −4, 4 — Denominator zero where x^2−16=0; no cancellation.
37. Find the coordinates of the hole for g(x)=(x^2−9)/(x^2−3x).
Answer
Hole at x=3. Factor to (x−3)(x+3)/[x(x−3)] → cancel (x−3); reduced form (x+3)/x. Evaluate at x=3 → y = 6/3 = 2. Hole at (3,2).
38. State the horizontal asymptote of h(x)=(7x^3−1)/(2x^3+4).
Answer
y = 7/2 — Equal degree; ratio 7/2.
39. Use long division to find the slant asymptote of r(x)=(x^3−2x+1)/(x^2+1).
Answer
y = x — Division quotient is x with lower-degree remainder.
40. Solve the inequality (x+3)/(x−1) < 0.
Answer
(−3, 1) — Zero at −3 (excluded), VA at 1 (excluded); negative between them.
41. Find all x-intercepts of f(x)=(x^2−4x)/(x^2+1).
Answer
x = 0 and x = 4 — Numerator x(x−4) = 0.
42. Evaluate lim (x→∞) (2x^2−5x+1)/(x^2+1).
Answer
2 — Leading-coefficient ratio 2/1.
43. Evaluate lim (x→−∞) (−x^3+1)/(2x^3+5).
Answer
−1/2 — Dominant terms (−x^3)/(2x^3) → −1/2.
44. Determine whether f(x)=(x^2−1)/(x^2+1) is even, odd, or neither.
Answer
Even — f(−x)=f(x).
45. Solve (x^2−1)/(x^2−4) > 0.
Answer
(−∞, −2) ∪ (−1, 1) ∪ (2, ∞) — Sign analysis at critical points x = −2, −1, 1, 2 with VAs excluded.
46. Find the y-intercept of g(x)=(3x−2)/(x^2−9).
Answer
(0, 2/9) — g(0) = −2/−9 = 2/9.
47. State the hole and simplified form for f(x)=(x^2+4x+3)/(x+3).
Answer
Hole at x = −3; simplified form x + 1 — Factor (x+1)(x+3) and cancel.
48. As x→∞, compute (5x^4+1)/(2x^4+7).
Answer
5/2 — Ratio of leading coefficients.
49. Identify all discontinuities of f(x)=(x^3−1)/(x^2−1).
Answer
Hole at x = 1; VA at x = −1 — Factor (x−1)(x^2+x+1)/[(x−1)(x+1)] and cancel (x−1).
50. Describe end behavior of r(x)=(x^4−2x)/(x^3+1).
Answer
Slant (linear) asymptote y = x — Degree difference 1; leading-term division yields x.