In Exercises 57–80, follow the seven steps to graph each rational function. f(x)=4x/(x−2)

Verified step by step guidance

1. Identify the domain of the function by determining the values of $x$ that make the denominator zero. For $f(x)\; =\; \backslash frac\{4x\}\{x\; -\; 2\}$, set $x\; -\; 2\; =\; 0$ and solve for $x$.
2. Find the vertical asymptote(s) by using the values of $x$ that are excluded from the domain. These occur where the denominator is zero, so the vertical asymptote is at $x\; =\; 2$.
3. Determine 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, which is $y\; =\; \backslash frac\{4\}\{1\}\; =\; 4$.
4. Calculate the x-intercept(s) by setting the numerator equal to zero and solving for $x$. For $4x\; =\; 0$, solve for $x$.
5. Calculate the y-intercept by evaluating $f(0)$. Substitute $x\; =\; 0$ into the function and simplify to find the point where the graph crosses the y-axis.

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 and behavior of rational functions is essential for graphing, as they often have restrictions like vertical asymptotes where the denominator is zero.

Asymptotes

Asymptotes are lines that the graph of a function approaches but never touches. For rational functions, vertical asymptotes occur where the denominator is zero, and horizontal or oblique asymptotes describe end behavior as x approaches infinity or negative infinity.

Graphing Steps for Rational Functions

Graphing rational functions involves a systematic approach: finding domain restrictions, identifying intercepts, determining asymptotes, analyzing end behavior, and plotting points. Following these steps ensures an accurate and complete graph of the function.