Question

show all work to identify the asymptotes and state the end behavior of the function $f(x) = \frac{4x}{x - 16}$.

Explanation:

Step1: Find Vertical Asymptote

To find the vertical asymptote of a rational function \( f(x)=\frac{N(x)}{D(x)} \), we set the denominator \( D(x) = 0 \) (provided the numerator \( N(x) \) is not zero at that point). For \( f(x)=\frac{4x}{x - 16} \), set \( x-16=0 \). Solving for \( x \), we get \( x = 16 \). Since the numerator \( 4x=4\times16 = 64
eq0 \) when \( x = 16 \), \( x = 16 \) is a vertical asymptote.

Step2: Find Horizontal Asymptote

For a rational function \( f(x)=\frac{N(x)}{D(x)} \) where the degree of \( N(x) \) (denoted as \( \text{deg}(N) \)) and the degree of \( D(x) \) (denoted as \( \text{deg}(D) \)) are equal, the horizontal asymptote is the ratio of the leading coefficients. Here, \( \text{deg}(N)=\text{deg}(D) = 1 \) (the leading term of \( N(x)=4x \) is \( 4x \) with coefficient 4, and the leading term of \( D(x)=x - 16 \) is \( x \) with coefficient 1). So the horizontal asymptote is \( y=\frac{4}{1}=4 \).

Step3: Analyze End Behavior

To find the end - behavior, we can consider the limit as \( x
ightarrow\infty \) and \( x
ightarrow-\infty \). We can rewrite \( f(x)=\frac{4x}{x - 16}=\frac{4x\div x}{(x - 16)\div x}=\frac{4}{1-\frac{16}{x}} \) (for \( x
eq0 \)).

• As \( x

ightarrow\infty \): \( \frac{16}{x}
ightarrow0 \), so \( f(x)=\frac{4}{1 - 0}=4 \). So \( \lim_{x
ightarrow\infty}f(x)=4 \).

• As \( x

ightarrow-\infty \): \( \frac{16}{x}
ightarrow0 \), so \( f(x)=\frac{4}{1-0} = 4 \). So \( \lim_{x
ightarrow-\infty}f(x)=4 \).

Also, we can analyze the behavior near the vertical asymptote \( x = 16 \):

• As \( x

ightarrow16^{+} \) (approaching 16 from the right), \( x - 16
ightarrow0^{+} \) and \( 4x
ightarrow64 \) (positive). So \( f(x)=\frac{4x}{x - 16}
ightarrow\frac{64}{0^{+}}
ightarrow\infty \).

• As \( x

ightarrow16^{-} \) (approaching 16 from the left), \( x - 16
ightarrow0^{-} \) and \( 4x
ightarrow64 \) (positive). So \( f(x)=\frac{4x}{x - 16}
ightarrow\frac{64}{0^{-}}
ightarrow-\infty \).

Answer:

• Vertical Asymptote: \( x = 16 \)
• Horizontal Asymptote: \( y=4 \)
• End - behavior: As \( x

ightarrow\pm\infty \), \( f(x)
ightarrow4 \); as \( x
ightarrow16^{+} \), \( f(x)
ightarrow\infty \); as \( x
ightarrow16^{-} \), \( f(x)
ightarrow-\infty \)