Question: Graph the following function. Determine the asymptotes, if any. List the domain and the…

Graph the following function. Determine the asymptotes, if any. List the domain and the x-intercepts and the y-intercepts. Check your work using a graphing calculator.

\[ f(x) = \frac{8}{x^2 + 9} \]

The domain of the function is \((-\infty, \infty)\).

(Type your answer in interval notation.)

What is the x-intercept of the function? Select the correct choice below, and if necessary, fill in the answer box to complete your choice.

A. The x-intercept(s) of the function is/are

(Type an ordered pair. Use a comma to separate answers as needed. Use integers or fractions for any numbers in the expression.)

B. The function has no x-intercepts.

What is the y-intercept of the function? Select the correct choice below, and if necessary, fill in the answer box to complete your choice.

A. The y-intercept of the function is

(Simplify your answer. Type an ordered pair.)

B. The function has no y-intercepts.

Solution

Step 1: Determine the domain of the function. \[ f(x) = \frac{8}{x^2 + 9} \] Since the denominator \( x^2 + 9 \) is always positive for all real numbers, the domain is all real numbers. \[ \text{Domain: } (-\infty, \infty) \] Step 2: Find the x-intercepts of the function. Set \( f(x) = 0 \): \[ \frac{8}{x^2 + 9} = 0 \] There are no solutions because the numerator is not zero. Therefore, the function has no x-intercepts. \[ \text{No x-intercepts} \] Step 3: Determine the y-intercept of the function. Set \( x = 0 \): \[ f(0) = \frac{8}{0^2 + 9} = \frac{8}{9} \] Thus, the y-intercept is at the point \( (0, \frac{8}{9}) \). \[ \text{y-intercept: } \left(0, \frac{8}{9}\right) \] Step 4: Determine the asymptotes of the function. Horizontal asymptote: \[ \lim_{x \to \pm\infty} \frac{8}{x^2 + 9} = 0 \] So, the horizontal asymptote is \( y = 0 \). Vertical asymptotes: Set the denominator equal to zero: \[ x^2 + 9 = 0 \] This equation has no real solutions, so there are no vertical asymptotes. \[ \text{Asymptotes: } y = 0 \text{ (horizontal)} \]