Question

select the correct choices that completes the sentence below. if the graph of a function f increases or decreases without bound as x approaches a, then the line x = a is of the graph of f. the equation of such a line for the graph of f(x) = \frac{1}{x + 2} is

Explanation:

Step1: Recall Vertical Asymptote Definition

A vertical asymptote is a vertical line \( x = a \) where the function \( f(x) \) increases or decreases without bound (tends to \( \pm\infty \)) as \( x \) approaches \( a \). So the first blank is "a vertical asymptote".

Step2: Find Vertical Asymptote for \( f(x)=\frac{1}{x + 2} \)

For a rational function \( \frac{g(x)}{h(x)} \), vertical asymptotes occur where \( h(x)=0 \) (and \( g(x)
eq0 \) at those points). For \( f(x)=\frac{1}{x + 2} \), set denominator \( x + 2 = 0 \). Solving \( x+2 = 0 \) gives \( x=-2 \). So the equation of the vertical asymptote is \( x=-2 \).

Answer:

First blank: a vertical asymptote; Second blank: \( x = -2 \)