Polynomial and Rational Functions — High School Pre-Calculus practice questions

Vertical asymptotes of f(x) = (x² − 1) / (x² − 4):

1. x = ±1
2. None
3. x = 1, x = −1, x = 2, x = −2
4. x = ±2

Answer: D — x = ±2

D) Vertical asymptotes occur where the denominator equals zero AND the numerator does not. x² − 4 = 0 → x = ±2; numerator at x = ±2 is 3, nonzero. A) Confused with zeros of numerator. C) Combined both. B) Misread that simplification removes them.

What is the formula for the nth term of an arithmetic sequence with first term a₁ and common difference d?

1. aₙ = a₁ · dⁿ⁻¹
2. aₙ = a₁ - n·d
3. aₙ = a₁ + n·d
4. aₙ = n·d
5. aₙ = a₁ + (n-1)·d (each term adds d to first; n-1 differences from start)

Answer: E — aₙ = a₁ + (n-1)·d (each term adds d to first; n-1 differences from start)

Arithmetic sequence formula: aₙ = a₁ + (n-1)d. n-1 steps from first term, each adding d. Different from geometric: aₙ = a₁·r^(n-1). Pre-calc sequences and series.

What is the value of cos(π/2)?

1. 1
2. Undefined
3. -1
4. √2/2
5. 0 (cos(90°) = 0)

Answer: E — 0 (cos(90°) = 0)

Special angle values: cos(0)=1, cos(π/6)=√3/2, cos(π/4)=√2/2, cos(π/3)=1/2, cos(π/2)=0, cos(π)=-1. Pre-Calc requires fluency on unit circle.

What is the inverse function of f(x) = e^x?

1. f⁻¹(x) = 1/eˣ
2. f⁻¹(x) = ln(x) (natural log is inverse of natural exponential)
3. f⁻¹(x) = x²
4. f⁻¹(x) = e^(-x)
5. f⁻¹(x) = log₁₀(x)

Answer: B — f⁻¹(x) = ln(x) (natural log is inverse of natural exponential)

Inverse pair: y = eˣ and y = ln(x). Their graphs are reflections across y = x. Properties: e^(ln x) = x; ln(eˣ) = x. Foundation of solving exponential equations using logs.

What feature does f(x) = (x² − 9)/(x − 3) have at x = 3?

1. Vertical asymptote
2. Jump discontinuity
3. Hole (removable discontinuity)
4. Smooth point

Answer: C — Hole (removable discontinuity)

C) Factor: (x − 3)(x + 3)/(x − 3) cancels, leaving x + 3 for x ≠ 3. So x = 3 is a hole, not a vertical asymptote. A) Misses the cancellation — a frequent trap. B) Jump requires different left/right finite limits. D) The original function is undefined at x = 3.

What is i, and what is i²?

i = √−1; i² = −1.

What is a hole in a rational function?

A point (a, b) removed from the graph because (x − a) cancels in numerator and denominator.

What is a slant (oblique) asymptote?

Occurs when degree of numerator = degree of denominator + 1; found by polynomial long division.

What is a horizontal asymptote? How do you find it?

A horizontal line y = L the function approaches as x → ±∞. Compare degrees of numerator and denominator: if deg(num) < deg(den), y = 0; equal, y = ratio of leading coefficients; greater, no horizontal asymptote.