Question

find the asymptotes and intercepts of the function, and graph the function.
$g(x)=\frac{x - 1}{x^2 - 2x - 24}$

\bigcirc c. the function has no vertical asymptotes.

find the x-intercepts. select the correct answer below and, if necessary, fill in the answer box to complete your choice.
\bigcirc a. the x-intercept(s) is/are \square
( type an integer or a fraction. use a comma to separate answers as needed )
\bigcirc b. there are no x-intercepts.

find the y-intercepts. select the correct answer below and, if necessary, fill in the answer box to complete your choice.
\bigcirc a. the y-intercept(s) is/are \square.
( type an integer or a fraction. use a comma to separate answers as needed. )
\bigcirc b. there are no y-intercepts.

Explanation:

Step1: Find x - intercepts

To find the x - intercepts, we set \(g(x)=0\). So we solve the equation \(\frac{x - 1}{x^{2}-2x - 24}=0\). A fraction is zero when its numerator is zero (and the denominator is not zero).
Set the numerator equal to zero: \(x - 1=0\), which gives \(x = 1\).
Now we check the denominator at \(x = 1\): \(x^{2}-2x - 24=(1)^{2}-2(1)-24=1 - 2 - 24=-25
eq0\). So \(x = 1\) is a valid x - intercept.

Step2: Find y - intercepts

To find the y - intercepts, we set \(x = 0\) in the function \(g(x)\).
\(g(0)=\frac{0 - 1}{0^{2}-2(0)-24}=\frac{-1}{-24}=\frac{1}{24}\).

Answer:

For x - intercepts:

A. The x - intercept(s) is/are \(1\)

For y - intercepts:

A. The y - intercept(s) is/are \(\frac{1}{24}\)