How to Find Intercepts in a Rational Function

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The intercepts of a function are the values of x when f(x) = 0 and the value of f(x) when x = 0, corresponding to the coordinate values of x and y where the graph of the function crosses the x- and y-axes. Find the y-intercept of a rational function as you would for any other type of function: plug in x = 0 and solve. Find the x-intercepts by factoring the numerator. Remember to exclude holes and vertical asymptotes when finding the intercepts.

  • Plug the value x = 0 into the rational function and determine the value of f(x) to find the y-intercept of the function. For example, plug x = 0 into the rational function f(x) = (x^2 - 3x + 2) / (x - 1) to get the value (0 - 0 + 2) / (0 - 1), which is equal to 2 / -1 or -2 (if the denominator is 0, there is a vertical asymptote or hole at x = 0 and therefore no y-intercept). The y-intercept of the function is y = -2.

  • Factor the numerator of the rational function completely. In the above example, factor the expression (x^2 - 3x + 2) into (x - 2)(x - 1).

  • Set the factors of the numerator equal to 0 and solve for the value of the variable to find the potential x-intercepts of the rational function. In the example, set the factors (x - 2) and (x - 1) equal to 0 to get the values x = 2 and x = 1.

  • Plug the values of x you found in Step 3 into the rational function to verify that they are x-intercepts. X-intercepts are values of x that make the function equal to 0. Plug x = 2 into the example function to get (2^2 - 6 + 2) / (2 - 1), which equals 0 / -1 or 0, so x = 2 is an x-intercept. Plug x = 1 into the function to get (1^2 - 3 + 2) / (1 - 1) to get 0 / 0, which means there is a hole at x = 1, so there is only one x-intercept, x = 2.

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