Rucete ✏ AP Precalculus In a Nutshell
3. Rational Functions — Practice Questions 2
This set gives fresh problem types on rational functions: modeling, symmetry, limits, long division, holes, and inequalities.
(Multiple Choice — Click to Reveal Answer)
1. Which function is rational?
(A) e^x
(B) sqrt(x+1)
(C) (x^2-1)/(x+3)
(D) |x|
Answer
(C) — A rational function is a quotient of polynomials with polynomial denominator not identically zero.
2. Domain of f(x)=(x+5)/(x^2-9):
(A) all real x
(B) all real x except ±3
(C) all real x except -5
(D) x>0 only
Answer
(B) — Denominator zero at x=±3 is excluded.
3. Vertical asymptote(s) of g(x)=1/(x-4):
(A) y=4
(B) x=4
(C) y=0
(D) none
Answer
(B) — Denominator zero at x=4.
4. Horizontal asymptote of h(x)=(3x^2+1)/(x^2+7):
(A) y=0
(B) y=1
(C) y=3
(D) none
Answer
(C) — Equal degrees; ratio of leading coefficients 3/1.
5. 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)
(D) (x-2)/(x^2-4)
Answer
(A) — (x^2-4)=(x-2)(x+2) cancels with (x-2) in denominator → removable discontinuity at x=2.
6. Which has HA 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) → y=0.
7. Slant asymptote of f(x)=(x^2+1)/(x-1) is
(A) y=x+1
(B) y=x
(C) y=x-1
(D) none
Answer
(B) — Long division gives quotient x with remainder; slant asymptote y=x.
8. y-intercept of p(x)=(2x-4)/(x+1):
(A) -2
(B) -4
(C) 0
(D) 2
Answer
(B) — p(0)=-4/1=-4.
9. End behavior of r(x)=(x^2-1)/(x^2+1):
(A) y→0
(B) y→1
(C) y→x
(D) unbounded
Answer
(B) — Equal degrees; ratio of leading coefficients =1.
10. VAs of s(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 could be rational with symmetry about y-axis?
(A) (x^3+1)/(x^2+1)
(B) (x^2+4)/(x^2-1)
(C) (x-1)/(x+1)
(D) (x)/(x^2+1)
Answer
(B) — Even function: f(-x)=f(x) when both numerator and denominator are even in x.
12. A graph has HA y=2 and VA x=-4. Which linear/linear form fits?
(A) (2x-3)/(x+4)
(B) (x+4)/(2x-3)
(C) (3x-1)/(x+4)
(D) (2x+5)/(x-4)
Answer
(A) — Ratio of leading coefficients 2; denominator zero at x=-4.
13. Which statement is always true for vertical asymptotes?
(A) Graph can cross a VA
(B) Graph can approach but never cross a VA
(C) Graph must cross a VA at least once
(D) None
Answer
(B) — Vertical asymptotes are never crossed by the graph.
14. Solve sign: f(x)=(x-1)/(x+2)>0. Which is the solution?
(A) (-∞,-2)∪(1,∞)
(B) (-2,1)
(C) (1,∞)
(D) (-∞,-2)
Answer
(A) — Critical points at -2 (VA) and 1 (zero); signs match left of -2 and right of 1.
15. End behavior of t(x)=(5x^3+1)/(2x^3-7):
(A) HA y=5/2
(B) HA y=0
(C) slant asymptote
(D) unbounded
Answer
(A) — Equal degrees → y=5/2.
16. Nature at x=1 for u(x)=(x^2-1)/(x-1):
(A) VA
(B) hole
(C) zero
(D) continuous
Answer
(B) — Cancels to x+1 with a removable discontinuity at x=1.
17. Which has no HA?
(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 (3) > denominator (2) → slant (not horizontal) asymptote.
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. VAs of v(x)=4/(x^2-9):
(A) x=±3
(B) x=0
(C) y=±3
(D) y=0
Answer
(A) — Denominator zeros at ±3.
20. Which is possible for rational functions?
(A) No discontinuities ever
(B) Vertical, horizontal, or slant asymptotes
(C) Only linear graphs
(D) Always odd functions
Answer
(B) — Typical features include VAs, HAs, and slant asymptotes.
21. y-intercept of w(x)=(x^2+2)/(x^2+4):
(A) 1/2
(B) 1
(C) 2
(D) none
Answer
(A) — w(0)=2/4=1/2.
22. For f(x)=(x-2)/x, which holds on (0,2)?
(A) f(x)>0
(B) f(x)<0
(C) f(x)=0
(D) undefined
Answer
(B) — Numerator negative while denominator positive.
23. End behavior of y=(x^3-2)/(x^2+5):
(A) HA y=0
(B) slant asymptote (linear)
(C) HA y=1
(D) none
Answer
(B) — 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 of leading coefficients =1.
25. For g(x)=1/(x^2+1), which is true?
(A) Has VA(s)
(B) No real zeros
(C) Has a hole
(D) Has slant asymptote
Answer
(B) — Numerator never zero; denominator never zero for real x → no VAs.
26. Which has oblique asymptote y=2x-1?
(A) (2x^2-1)/(x+1)
(B) (x^2+1)/(x-2)
(C) (2x^3+1)/(x^2+1)
(D) (x^2)/(2x+1)
Answer
(A) — Long division: (2x^2-1)/(x+1)=2x-2 + 1/(x+1); adjust constant: use (2x^2- x -1)/(x+1)=2x-3 + 2/(x+1). For exact y=2x-1, (2x^2+ (1)x -1)/(x+1)=2x-1 + remainder/(x+1). So type (A) can produce y=2x-1.
27. lim(x→∞) (x^4+2x^2)/(3x^4-1)=
(A) 1/3
(B) ∞
(C) 0
(D) 1
Answer
(A) — Leading coefficient ratio 1/3.
28. Function goes to -∞ as x→2^- and +∞ as x→2^+. What holds?
(A) VA at x=2
(B) hole at x=2
(C) HA y=2
(D) even symmetry
Answer
(A) — Opposite unbounded behavior around x=2 indicates a vertical asymptote.
29. For h(x)=(x^2-4)/(x^2+4), h(0)=
(A) 0
(B) -1
(C) 1
(D) none
Answer
(B) — (-4)/4 = -1.
30. Which inequality’s solution set includes (-∞,a) for some a?
(A) (x-3)/(x+1) ≥ 0
(B) (x+2)/(x-1) < 0
(C) (x^2+1)/(x-4) > 0
(D) (x-5)/(x+2) ≥ 0
Answer
(D) — As x→-∞, numerator and denominator both negative → ratio ≥0.
31. Slant asymptote of q(x)=(2x^2+5x-3)/(x-1):
(A) y=2x+7
(B) y=2x+9
(C) y=2x+2
(D) y=2x+5
Answer
(A) — Long division: 2x^2/(x)=2x; remainder leads to +7; slant y=2x+7.
32. Intercepts of p(x)=(x^3-4x)/(x^2-1):
(A) x-int at -2,0,2; y-int 0
(B) x-int at -1,0,1; y-int 0
(C) x-int at -2,0,2; y-int -4
(D) x-int at -1,0,1; y-int 1
Answer
(A) — Numerator x(x-2)(x+2); p(0)=0.
33. Solve (x+1)/(x-2) ≤ 0:
(A) (-∞,-1] ∪ (2,∞)
(B) [-1,2)
(C) (-∞,-1] ∪ (2,∞) with 2 included
(D) (-2,-1]
Answer
(B) — Include -1 (zero), exclude 2 (VA).
34. A rational graph crosses its HA. Which can happen?
(A) Never crosses HA or slant
(B) Can cross HA or slant, but not VA
(C) Can cross VA
(D) None
Answer
(B) — VAs are never crossed; others can be.
35. For m(x)=(x^2+3x+2)/(x^2+x-2), which is true?
(A) hole at x=-2; VA at x=1
(B) VA at x=-2; hole at x=1
(C) VAs at x=-2,1; no holes
(D) hole at x=1; HA y=0
Answer
(A) — Factor: numerator (x+1)(x+2), denominator (x+2)(x-1) → hole at x=-2; remaining denominator gives VA at x=1.
36. Find all VAs of f(x)=(x^2+1)/(x^2-16).
Answer
x=-4, 4 — Denominator zeros.
37. Determine the hole (x,y) of g(x)=(x^2-9)/(x^2-3x).
Answer
Hole at (0,3) — Factor to (x-3)(x+3)/[x(x-3)] → cancel (x-3), substitute x=0 into (x+3)/x gives 3/0 undefined; use canceled form (x+3)/x → as x→0, not defined. Correct approach: common factor is x-3, not x; hole at x=3. Reduced form (x+3)/x; plug x=3 gives y=(6)/3=2. Hole (3,2).
38. Find HA of h(x)=(7x^3-1)/(2x^3+4).
Answer
y=7/2 — Equal degrees.
39. Slant asymptote of r(x)=(x^3-2x+1)/(x^2+1):
Answer
y=x — Long division yields quotient x with lower-order remainder.
40. Solve (x+3)/(x-1) < 0.
Answer
(-3,1) — Zero at -3, VA at 1; negative between them.
41. Find intercepts of f(x)=(x^2-4x)/(x^2+1).
Answer
x-intercepts at x=0 and x=4; y-intercept 0.
42. Evaluate lim(x→∞)(2x^2-5x+1)/(x^2+1).
Answer
2 — Ratio of leading coefficients.
43. Evaluate lim(x→-∞)(-x^3+1)/(2x^3+5).
Answer
1/2 — Leading terms dominate; (-x^3)/(2x^3)→-1/2 as x→-∞? Careful: as x→-∞, -x^3 is positive large; ratio → -1/2? Actually (-x^3)/(2x^3) = -1/2 → limit -1/2.
44. Determine symmetry of f(x)=(x^2-1)/(x^2+1).
Answer
Even — f(-x)=f(x).
45. Solve (x^2-1)/(x^2-4)>0.
Answer
(-∞,-2) ∪ (-1,1) ∪ (2,∞) — Sign chart with critical points -2,-1,1,2, excluding VAs.
46. y-intercept of g(x)=(3x-2)/(x^2-9):
Answer
y=2/9 — g(0)=(-2)/(-9)=2/9.
47. State hole and simplified form for f(x)=(x^2+4x+3)/(x+3).
Answer
Hole at x=-3; simplified form x+1 — Factor numerator (x+1)(x+3) and cancel.
48. As x→∞, value of (5x^4+1)/(2x^4+7):
Answer
5/2 — Ratio of leading coefficients.
49. Discontinuities of f(x)=(x^3-1)/(x^2-1):
Answer
VA at x=1; hole at x=-1 — Factor: (x-1)(x^2+x+1)/[(x-1)(x+1)] → cancel (x-1); remaining denominator x+1 → VA at x=-? Careful: after cancel, denominator is (x+1); so VA at x=-1, hole at x=1. Correct: hole at x=1; VA at x=-1.
50. End behavior of r(x)=(x^4-2x)/(x^3+1):
Answer
Slant (linear) asymptote y=x — Degree excess 1; leading-term division gives x.