Rational Functions: Learn It 5

Finding Intercepts of Rational Functions

We can always find any horizontal intercepts of the graph of a function by setting the output equal to zero. We can always find any vertical intercept by setting the input equal to zero.

In a rational function of the form [latex]r(x)=\dfrac{P(x)}{Q(x)}[/latex],

  • Find the vertical intercept (the [latex]y[/latex]-intercept) by evaluating [latex]r(0)[/latex]. That is, replace all the input variables with [latex]0[/latex] and calculate the result.
  • Find the horizontal intercept(s) (the [latex]x[/latex]-intercepts) by solving [latex]r(x)=0[/latex]. Since the function is undefined where the denominator equals zero], set the numerator equal to zero to find the horizontal intercepts of the function.
  • Note that the graph of a rational function may not possess a vertical- or horizontal-intercept.

intercepts of rational functions

A rational function will have a [latex]y[/latex]-intercept when the input is zero, if the function is defined at zero. A rational function will not have a [latex]y[/latex]-intercept if the function is not defined at zero.
[latex]\\[/latex]

Likewise, a rational function will have [latex]x[/latex]-intercepts at the inputs that cause the output to be zero. Since a fraction is only equal to zero when the numerator is zero, [latex]x[/latex]-intercepts can only occur when the numerator of the rational function is equal to zero.

Find the intercepts of [latex]f\left(x\right)=\dfrac{\left(x - 2\right)\left(x+3\right)}{\left(x - 1\right)\left(x+2\right)\left(x - 5\right)}[/latex].