Graph each rational function. ƒ(x)=(x+2)/(x-3)

Verified step by step guidance

1. Identify the rational function given: \(f(x) = \frac{x+2}{x-3}\).
2. Determine the vertical asymptote by finding the values of \(x\) that make the denominator zero. Set \(x - 3 = 0\) and solve for \(x\).
3. Find the horizontal asymptote by comparing the degrees of the numerator and denominator. Since both numerator and denominator are degree 1, the horizontal asymptote is the ratio of the leading coefficients.
4. Calculate the \(x\)-intercept by setting the numerator equal to zero and solving for \(x\): \(x + 2 = 0\).
5. Calculate the \(y\)-intercept by evaluating \(f(0)\), which means substituting \(x = 0\) into the function.

Key Concepts

Rational Functions

A rational function is a ratio of two polynomials, expressed as f(x) = P(x)/Q(x), where Q(x) ≠ 0. Understanding the form helps identify domain restrictions and behavior, such as vertical asymptotes where the denominator is zero.

Domain and Vertical Asymptotes

The domain of a rational function excludes values that make the denominator zero. These values often correspond to vertical asymptotes, where the function approaches infinity or negative infinity, indicating points of discontinuity on the graph.

Graphing Rational Functions

Graphing involves identifying intercepts, asymptotes (vertical, horizontal, or oblique), and behavior near these lines. For f(x) = (x+2)/(x-3), find zeros, vertical asymptotes at x=3, and horizontal asymptotes by comparing degrees of numerator and denominator.