Question

the graph of function f is shown. which statement correctly describes the graph of ( g(x) = f(x - 4) )?

a. function g has the same vertical asymptote as f and a horizontal asymptote at ( y = 2 ).

b. function g has a horizontal asymptote at ( y = 2 ) and a vertical asymptote at ( x = 5 ).

c. function g has the same vertical and horizontal asymptotes as function f.

d. function g has the same horizontal asymptote as f and a vertical asymptote at ( x = 5 ).

Explanation:

Step1: Identify asymptotes of $f(x)$

From the graph:

• Horizontal asymptote of $f(x)$: $y=2$
• Vertical asymptote of $f(x)$: $x=1$

Step2: Analyze transformation for $g(x)$

For $g(x)=f(x-4)$, this is a horizontal shift right by 4 units.

• Horizontal asymptotes are unchanged under horizontal shifts: $y=2$ (same as $f(x)$)
• Vertical asymptote shifts right by 4: $x=1+4=5$

Step3: Match with options

Compare the result to the given choices.

Answer:

D. Function $g$ has the same horizontal asymptote as $f$ and a vertical asymptote at $x = 5$