AP Precalculus Rational Functions

Question 1

Which line is the horizontal asymptote of f(x) = (3x² − 5x + 1)/(x² + 4)?

Accepted Answer

y = 3

Answer

The numerator and denominator both have degree 2, so the horizontal asymptote is the ratio of leading coefficients, 3/1 = 3.

Question 2

Consider f(x) = (x − 2)/((x − 2)(x + 3)). Which statement correctly describes the graph of f?

Accepted Answer

A vertical asymptote at x = −3 and a hole at x = 2

Answer

The factor (x − 2) cancels, which leaves a hole at x = 2. The factor (x + 3) remains in the denominator, which gives a vertical asymptote at x = −3.

Question 3

What are the zeros of f(x) = (x² − 4)/(x² − x − 6)?

Accepted Answer

x = 2 only

Answer

Factoring gives f(x) = (x − 2)(x + 2)/((x − 3)(x + 2)). The factor (x + 2) cancels, so x = −2 is a hole, not a zero. The only zero is x = 2.

Question 4

As x → ∞ and as x → −∞, the graph of f(x) = (x² + 1)/(x − 1) approaches which line?

Accepted Answer

y = x + 1

Answer

Polynomial division gives x² + 1 = (x − 1)(x + 1) + 2, so f(x) = x + 1 + 2/(x − 1). The remainder term 2/(x − 1) approaches 0, which leaves the slant line y = x + 1.

Question 5

Which expression is equivalent to (2x² + 3x − 5)/(x − 1) for all x ≠ 1?

Accepted Answer

2x + 5

Answer

The numerator factors as (2x + 5)(x − 1), so the factor (x − 1) cancels and leaves 2x + 5 for x ≠ 1.

AP Precalculus Practice Test

AP Precalculus Practice Test

AP Precalculus Rational Functions