Question

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let $f(x) = \frac{x + 1}{x + 4}$. the following is the graph of $f(x)$.
graph of the function $f(x)$ with axes and grid
a) what is the degree of the numerator of $f(x)$?
b) what is the degree of the denominator of $f(x)$?
c) enter the equation of the horizontal asymptote of $f(x)$.
d) enter the equations of all vertical asymptotes of $f(x)$. separate multiple equations with a comma.

Explanation:

Part (a)

Step1: Identify numerator

The numerator of \( f(x)=\frac{x + 1}{x + 4} \) is \( x + 1 \).

Step2: Determine degree

The highest power of \( x \) in \( x + 1 \) is \( 1 \), so the degree is \( 1 \).

Part (b)

Step1: Identify denominator

The denominator of \( f(x)=\frac{x + 1}{x + 4} \) is \( x + 4 \).

Step2: Determine degree

The highest power of \( x \) in \( x + 4 \) is \( 1 \), so the degree is \( 1 \).

Part (c)

Step1: Recall horizontal asymptote rule

For a rational function \( f(x)=\frac{N(x)}{D(x)} \), if the degrees of \( N(x) \) and \( D(x) \) are equal, the horizontal asymptote is \( y=\frac{\text{leading coefficient of }N(x)}{\text{leading coefficient of }D(x)} \).

Step2: Apply rule

Here, degrees of numerator and denominator are both \( 1 \). Leading coefficient of numerator (\( x + 1 \)) is \( 1 \), leading coefficient of denominator (\( x + 4 \)) is \( 1 \). So \( y=\frac{1}{1}=1 \).

Part (d)

Answer:

1