Question

which graph represents the function $f(x)=\frac{2}{x - 1}+4$?

Explanation:

Step1: Identify Vertical Asymptote

For the function \( f(x)=\frac{2}{x - 1}+4 \), the vertical asymptote occurs where the denominator is zero, i.e., \( x - 1 = 0\Rightarrow x = 1 \). So the graph should have a vertical asymptote at \( x = 1 \).

Step2: Identify Horizontal Asymptote

For rational functions of the form \( \frac{a}{x - h}+k \), the horizontal asymptote is \( y = k \). Here, \( k = 4 \), so the horizontal asymptote is \( y = 4 \).

Step3: Analyze Behavior Around Asymptotes

• For \( x>1 \) (right of \( x = 1 \)), as \( x

ightarrow1^+ \), \( \frac{2}{x - 1}
ightarrow+\infty \), so \( f(x)
ightarrow+\infty \); as \( x
ightarrow+\infty \), \( \frac{2}{x - 1}
ightarrow0 \), so \( f(x)
ightarrow4 \).

• For \( x<1 \) (left of \( x = 1 \)), as \( x

ightarrow1^- \), \( \frac{2}{x - 1}
ightarrow-\infty \), so \( f(x)
ightarrow-\infty \); as \( x
ightarrow-\infty \), \( \frac{2}{x - 1}
ightarrow0 \), so \( f(x)
ightarrow4 \).

We check the given graph (the first one shown) has vertical asymptote at \( x = 1 \), horizontal asymptote at \( y = 4 \), and the behavior matches the function's properties (right side approaches \( +\infty \) near \( x = 1 \) and approaches \( 4 \) as \( x
ightarrow\pm\infty \), left side approaches \( -\infty \) near \( x = 1 \) and approaches \( 4 \) as \( x
ightarrow-\infty \)).

Answer:

The graph (the one with vertical asymptote \( x = 1 \) and horizontal asymptote \( y = 4 \), showing the correct behavior around \( x = 1 \) and \( y = 4 \)) represents the function \( f(x)=\frac{2}{x - 1}+4 \). (Assuming the first graph in the provided image has these properties, so the answer is the graph with vertical asymptote \( x = 1 \) and horizontal asymptote \( y = 4 \) as analyzed.)