Final answer:
To find the x-intercepts of a rational function, set the numerator equal to zero and solve for x, while ensuring those solutions do not make the denominator zero.
Step-by-step explanation:
To solve for x-intercepts in rational functions, you determine where the function equals zero. The x-intercept(s) occurs where the numerator is equal to zero, assuming that the denominator is not also zero at these points. A rational function is a ratio of two polynomials, expressed as f(x) = P(x)/Q(x), where P(x) and Q(x) are polynomials and Q(x) ≠ 0. The steps to find the x-intercepts of the function are:
- Set the numerator equal to zero (P(x) = 0).
- Solve the resulting equation for x.
- Check to ensure that these solutions do not make the denominator zero, as those points would be undefined, not x-intercepts.
For example, consider the rational function f(x) = (x - 3)/(x^2 + x - 6). To find the x-intercept, set x - 3 = 0, which gives x = 3. You need to ensure that x = 3 does not also make the denominator zero. Since the denominator at x = 3 is 3^2 + 3 - 6 = 6, which is not zero, x = 3 is the x-intercept of the function.