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4.3 The X-intercepts

The X-intercepts for this function would be where the output, or y-value, equals zero. A rational function can be considered a fraction, and a fraction is equal to zero when the numerator is equal to zero. For our rational function example this happens when the polynomial in the numerator is equal to zero, and this will happen at the roots of this polynomial. The roots of the numerator polynomial are x = -5 and x = 4. That is, when x takes on either of these two values the numerator becomes zero, and the output of the function, or y-value, also becomes zero.

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So, the x-intercepts for this rational function are x = -5 and x = 4. Notice that the function is defined at these two values. (It is only not defined at x = -3 and x = 6.) That makes these true x-intercepts. If the function was not defined at x = 4 because 4 was a root of the denominator polynomial, (which it is not in our example here), then x = 4 would not be an x-intercept even though it made the numerator equal to zero. One cannot have an x-intercept for a function at a point where the function does not exist!

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