Questions: Use the a, b, and c-sliders to graph the function f(x)=3/(x+2)+1. Complete parts 1 through 3 below. Use the interactive figure to find your answer. Use the left and right arrow keys to move along a slider as needed. Click here to launch the interactive figure. Part 1: What is the domain of the function? The domain is (-∞,-2) ∪(-2, ∞). (Type your answer in interval notation.) Part 2: What is the equation of the vertical asymptote? A. y=x B. y=1 C. x=-y D. x=-2

Solution Steps

To solve the given parts of the question, we need to analyze the function \( f(x) = \frac{3}{x+2} + 1 \).

1. Domain of the function: The domain of a function is the set of all possible input values (x-values) for which the function is defined. For the given function, the denominator \( x+2 \) cannot be zero, as division by zero is undefined. Therefore, the domain excludes \( x = -2 \).

2. Equation of the vertical asymptote: A vertical asymptote occurs where the function approaches infinity, which happens when the denominator is zero. For the function \( f(x) = \frac{3}{x+2} + 1 \), the vertical asymptote is at \( x = -2 \).

The function given is \( f(x) = \frac{3}{x+2} + 1 \). To find the domain, we need to identify the values of \( x \) for which the function is defined. The function is undefined where the denominator is zero. Therefore, we solve the equation:

\[ x + 2 = 0 \]

Solving for \( x \), we find:

\[ x = -2 \]

Thus, the domain of the function is all real numbers except \( x = -2 \). In interval notation, the domain is:

\[ (-\infty, -2) \cup (-2, \infty) \]

A vertical asymptote occurs where the function approaches infinity, which is at the point where the denominator is zero. From the previous step, we found that the denominator is zero at \( x = -2 \). Therefore, the equation of the vertical asymptote is:

\[ x = -2 \]

Final Answer