Question: List the domain and the x- and y-intercepts of the following function. Graph the function. Be sure…

List the domain and the x- and y-intercepts of the following function. Graph the function. Be sure to label all the asymptotes.

\[ f(x) = \frac{x^2 - 7x + 10}{x + 2} \]

What is the domain of the function?

(Type your answer in interval notation.)

Solution

Let’s solve the problem step by step. To find the domain of the function \( f(x) = \frac{x^2 - 7x + 10}{x + 2} \): The denominator cannot be zero, so: \[ x + 2 \neq 0 & \] \[ x \neq -2 & \] Therefore, the domain in interval notation is: \[ (-\infty, -2) \cup (-2, \infty) \] Next, let’s find the x-intercepts by setting \( f(x) = 0 \): \[ \frac{x^2 - 7x + 10}{x + 2} = 0 & \] \[ x^2 - 7x + 10 = 0 & \] Factoring the quadratic equation: \[ (x - 2)(x - 5) = 0 & \] So, the x-intercepts are at: \[ x = 2 \quad \text{and} \quad x = 5 & \] Now, find the y-intercept by evaluating \( f(0) \): \[ f(0) = \frac{0^2 - 7(0) + 10}{0 + 2} = \frac{10}{2} = 5 & \] So, the y-intercept is at: \[ y = 5 & \] Finally, let’s identify the asymptotes: - Vertical Asymptote: Occurs where the denominator is zero. \[ x = -2 & \] - Oblique (Slant) Asymptote: Since the degree of the numerator is one higher than the denominator, perform polynomial division. Dividing \( x^2 - 7x + 10 \) by \( x + 2 \): \[ x^2 - 7x + 10 = (x + 2)(x - 9) + 28 & \] So, the oblique asymptote is: \[ y = x - 9 & \] Summary: - Domain: \( (-\infty, -2) \cup (-2, \infty) \) - x-intercepts: \( x = 2 \) and \( x = 5 \) - y-intercept: \( y = 5 \) - Vertical Asymptote: \( x = -2 \) - Oblique Asymptote: \( y = x - 9 \)