Rational Functions Flashcards

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

Answer: y = 3

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

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

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

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.

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

Answer: x = 2 only

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.

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

Answer: y = x + 1

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.

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

Answer: 2x + 5

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

For f(x) = (x + 1)/(x − 4), what happens to f(x) as x approaches 4 from the left (x → 4⁻)?

Answer: f(x) decreases without bound (→ −∞)

Near x = 4 from the left, the numerator is close to 5 (positive) and the denominator is a small negative number. The quotient is a large negative number, so f(x) → −∞.

What is the end behavior of f(x) = (x³ − 1)/(x² + 2)?

Answer: As x → ∞, f(x) → ∞, and as x → −∞, f(x) → −∞

The numerator has a higher degree than the denominator. For large |x|, f(x) behaves like x³/x² = x, which goes to ∞ as x → ∞ and to −∞ as x → −∞.

A rational function has zeros at x = 1 and x = −3, a vertical asymptote at x = 2, and a horizontal asymptote at y = 4. Which function fits all of these?

Answer: f(x) = 4(x − 1)(x + 3)/(x − 2)²

The numerator's zeros are 1 and −3, and the denominator's zero is 2. Both have degree 2, and the ratio of leading coefficients is 4/1 = 4, which gives the horizontal asymptote y = 4.

Where is the hole in the graph of f(x) = (x² + x − 6)/(x² − 4)?

Answer: (2, 5/4)

Factoring gives f(x) = (x + 3)(x − 2)/((x − 2)(x + 2)). The factor (x − 2) cancels, so the hole is at x = 2. The simplified function (x + 3)/(x + 2) gives y = 5/4 there.

Which expression is equivalent to (x² + 5x + 7)/(x + 2) for x ≠ −2 and shows its slant asymptote?

Answer: x + 3 + 1/(x + 2)

Dividing gives quotient x + 3 and remainder 1, because (x + 3)(x + 2) = x² + 5x + 6. So the expression equals x + 3 + 1/(x + 2).