Graph each rational function. See Examples 5–9. ƒ(x)=[(x+6)(x-2)]/[(x+3)(x-4)]

A rational function is a function that can be expressed as the ratio of two polynomials. The general form is f(x) = P(x)/Q(x), where P(x) and Q(x) are polynomials. Understanding the behavior of rational functions involves analyzing their asymptotes, intercepts, and the domain, which is restricted by the values that make the denominator zero.

Asymptotes are lines that the graph of a function approaches but never touches. For rational functions, vertical asymptotes occur at values of x that make the denominator zero, while horizontal asymptotes describe the behavior of the function as x approaches infinity. Identifying these asymptotes is crucial for sketching the graph accurately.

Intercepts are points where the graph of a function crosses the axes. The x-intercepts occur when f(x) = 0, which happens when the numerator is zero, while the y-intercept is found by evaluating f(0). Knowing the intercepts helps in plotting the graph and understanding the function's behavior in relation to the axes.