How do you find the domain of a rational function?


Quick Answer

A rational expression is a fraction. As with all fractions, the denominator of the expression cannot be equal to zero. The domain of an expression is all of the possible values for x. Finding all the values for x that equal zero shows which x values are restricted and cannot be part of the domain.

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Full Answer

  1. Determine which values for x generate a zero in the denominator

    The denominator of a rational expression cannot equal zero. In a simple rational expression, such as 3/x, the only value for x that is not allowed is x = 0, so the domain is expressed as "all x not equal to zero." If there is no x in the denominator, the domain is unrestricted, or "all x."

  2. Solve for x by setting the expression equal to zero

    If the denominator is a polynomial, the restricted values for x can be found by setting the denominator equal to zero and solving for x. For example, consider the rational expression 1/x^2 - 4. If x^2 - 4 = 0, then x^2 = 4 and x = 2 or -2, so the domain is "all x not equal to 2 or -2."

  3. Factor the polynomial

    If the denominator is a quadratic polynomial, factor the polynomial to find the restricted values for x. For example, consider the rational expression x + 2/x^2 + 2x - 15. The x in the numerator has no effect on the value of the denominator, so the only restricted values of x are only those that make x^2 + 2x - 15 = 0. Factoring x^2 + 2x - 15 yields the binomials (x + 5) and (x - 3), so the only values of x that will make the denominator zero are -5 and 3, and the domain is "all values of x not equal to 3 or -5."

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