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.

\[ f(x) = \frac{5}{x^2 + 3} \]

The domain of the function is \(\boxed{\quad}\).

(Type your answer in interval notation.)

Solution

To solve the given function \( f(x) = \frac{5}{x^2 + 3} \), let’s address each part of the problem step by step. 1. Graph the Function The function \( f(x) = \frac{5}{x^2 + 3} \) is a rational function where the denominator \( x^2 + 3 \) is always positive. As \( x \) approaches infinity or negative infinity, \( f(x) \) approaches zero. The graph will be a smooth curve that is always above the x-axis and approaches the horizontal asymptote \( y = 0 \). 2. Determine the Asymptotes Vertical asymptotes occur where the denominator is zero. \[ x^2 + 3 = 0 \] \[ x^2 = -3 \] Since there are no real solutions to this equation, there are no vertical asymptotes. Horizontal asymptotes are determined by the behavior of the function as \( x \) approaches infinity or negative infinity. \[ \lim_{{x \to \pm\infty}} \frac{5}{x^2 + 3} = 0 \] \[ y = 0 \] There is a horizontal asymptote at \( y = 0 \). 3. List the Domain The domain of the function includes all real numbers except where the denominator is zero. Since \( x^2 + 3 \) is never zero for real \( x \), the domain is: \[ (-\infty, \infty) \] 4. Find the x-intercepts and y-intercepts y-intercept: Set \( x = 0 \): \[ f(0) = \frac{5}{0^2 + 3} = \frac{5}{3} \] The y-intercept is \( \left(0, \frac{5}{3}\right) \). x-intercepts: Set \( f(x) = 0 \): \[ \frac{5}{x^2 + 3} = 0 \] \[ 5 = 0 \] This equation has no real solutions, so there are no x-intercepts. 5. Check the Work Using a Graphing Calculator To verify the results: - Plot the function \( f(x) = \frac{5}{x^2 + 3} \) on a graphing calculator. - Observe that the graph approaches \( y = 0 \) as \( x \) moves away from the origin. - Confirm that the only intercept is the y-intercept at \( \left(0, \frac{5}{3}\right) \).