Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. ƒ(x)=(2x+6)/(x-4)

Verified step by step guidance

1. Identify the rational function given: \(f(x) = \frac{2x + 6}{x - 4}\).
2. Find the vertical asymptotes by setting the denominator equal to zero and solving for \(x\): solve \(x - 4 = 0\) to find the vertical asymptote(s).
3. Determine the horizontal or oblique asymptotes by comparing the degrees of the numerator and denominator polynomials. Here, both numerator and denominator are degree 1.
4. Since the degrees of numerator and denominator are equal, find the horizontal asymptote by dividing the leading coefficients: the horizontal asymptote is \(y = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}\).
5. Summarize the asymptotes: vertical asymptote(s) from step 2 and horizontal asymptote from step 4. There is no oblique asymptote because the degrees are equal.

Key Concepts

Vertical Asymptotes

Vertical asymptotes occur where the denominator of a rational function equals zero, causing the function to approach infinity or negative infinity. For ƒ(x) = (2x+6)/(x-4), setting the denominator x-4 = 0 gives x = 4, indicating a vertical asymptote at x = 4.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as x approaches infinity or negative infinity. They depend on the degrees of the numerator and denominator polynomials. If degrees are equal, the horizontal asymptote is the ratio of leading coefficients; for ƒ(x), both are degree 1, so y = 2/1 = 2.

Oblique (Slant) Asymptotes

Oblique asymptotes occur when the degree of the numerator is exactly one more than the denominator. They are found by performing polynomial division. Since ƒ(x) has numerator and denominator of the same degree, there is no oblique asymptote in this case.