Graph each rational function. ƒ(x)=(6-3x)/(4-x)

Verified step by step guidance

1. Identify the rational function given: \(f(x) = \frac{6 - 3x}{4 - x}\).
2. Determine the domain by finding values of \(x\) that make the denominator zero. Set \$4 - x = 0\( and solve for \)x$ to find any vertical asymptotes.
3. Find the horizontal asymptote by comparing the degrees of the numerator and denominator. Since both numerator and denominator are linear (degree 1), divide the leading coefficients to find the horizontal asymptote.
4. Calculate the \(x\)-intercept by setting the numerator equal to zero: \$6 - 3x = 0\(, then solve for \)x\(. Calculate the \)y\(-intercept by evaluating \)f(0)$.
5. Plot the vertical and horizontal asymptotes on the graph, mark the intercepts, and sketch the curve considering the behavior near the asymptotes and intercepts.

Key Concepts

Rational Functions

A rational function is a ratio of two polynomials, expressed as f(x) = P(x)/Q(x). Understanding its domain, where the denominator Q(x) ≠ 0, is essential to avoid undefined values and to analyze the function's behavior.

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, often found by comparing degrees of numerator and denominator.

Graphing Rational Functions

Graphing involves plotting intercepts, asymptotes, and analyzing the function's behavior near these lines. Identifying points of discontinuity and understanding how the function approaches asymptotes help create an accurate graph.