Graph each rational function. ƒ(x)=(x+1)/(x-4)

Verified step by step guidance

1. Identify the rational function given: \(f(x) = \frac{x+1}{x-4}\).
2. Determine the vertical asymptote by finding the values of \(x\) that make the denominator zero. Set \(x - 4 = 0\) 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 \(x\), i.e., solve \(x + 1 = 0\).
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 domain restrictions and behavior of these functions is essential, as they often have asymptotes and discontinuities where the denominator is zero.

Vertical and Horizontal Asymptotes

Vertical asymptotes occur where the denominator equals zero, indicating values excluded from the domain. Horizontal asymptotes describe the end behavior of the function as x approaches infinity or negative infinity, determined by comparing the degrees of the numerator and denominator polynomials.

Graphing Rational Functions

Graphing involves identifying intercepts, asymptotes, and behavior near these lines. Plot key points, analyze limits near asymptotes, and use symmetry or transformations to sketch the curve accurately, providing a visual understanding of the function's behavior.