Question

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

Explanation:

Step1: Identify Vertical Asymptote

The vertical asymptote of a rational function \( \frac{a}{x - h}+k \) occurs where the denominator is zero. For \( f(x)=\frac{2}{x - 1}+4 \), set \( x - 1 = 0 \), so \( x = 1 \).

Step2: Identify Horizontal Asymptote

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

Step3: Analyze Transformations

The parent function is \( y=\frac{2}{x} \). The function \( f(x)=\frac{2}{x - 1}+4 \) is a horizontal shift of the parent function 1 unit to the right and a vertical shift 4 units up.

Step4: Key Points or Behavior

• As \( x

ightarrow1^{+} \), \( \frac{2}{x - 1}
ightarrow+\infty \), so \( f(x)
ightarrow+\infty + 4=+\infty \).

• As \( x

ightarrow1^{-} \), \( \frac{2}{x - 1}
ightarrow-\infty \), so \( f(x)
ightarrow-\infty + 4=-\infty \).

• As \( x

ightarrow\pm\infty \), \( \frac{2}{x - 1}
ightarrow0 \), so \( f(x)
ightarrow0 + 4 = 4 \).

To determine the correct graph, we look for a hyperbola with vertical asymptote \( x = 1 \) and horizontal asymptote \( y = 4 \), shifted 1 unit right and 4 units up from the parent \( y=\frac{2}{x} \).

(Note: Since the actual graphs are not provided here, but the process to identify the correct graph is based on vertical asymptote \( x = 1 \), horizontal asymptote \( y = 4 \) and the transformation from the parent rational function.)

Answer:

The graph should have a vertical asymptote at \( x = 1 \), a horizontal asymptote at \( y = 4 \), be a transformation (shifted 1 unit right and 4 units up) of the function \( y=\frac{2}{x} \), with the described end - behavior and asymptotes. If we were to match among given graphs, the one with these asymptotes and the correct shape (hyperbola) due to the rational function nature would be the correct graph.