Question

find the asymptotes and intercepts of the function, and graph the function.
$g(x)=\frac{x - 1}{x^2 - x - 30}$

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.)

Explanation:

Step1: Find x - intercepts

To find the x - intercepts, we set \(y = g(x)=0\). So we solve the equation \(\frac{x - 1}{x^{2}-x - 30}=0\). 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 check the denominator at \(x = 1\): \(x^{2}-x - 30=1^{2}-1 - 30=1 - 1-30=- 30
eq0\). So \(x = 1\) is a valid x - intercept.

Step2: Find y - intercepts

To find the y - intercepts, we set \(x = 0\) in the function \(g(x)\).
Substitute \(x = 0\) into \(g(x)=\frac{x - 1}{x^{2}-x - 30}\): \(g(0)=\frac{0 - 1}{0^{2}-0 - 30}=\frac{-1}{-30}=\frac{1}{30}\).

Answer:

For x - intercepts:

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

For y - intercepts:

A. The y - intercept(s) is/are \(\frac{1}{30}\).