Sign of a rational function

The zeros of the rational function are determined by the zeros of the numerator. At the zeros of the denominator, the function has a discontinuity point, and the function is not defined.

Example 1

Find the zeros of the function f.

The zero point of the function f is x = 2 and the function is not defined when x = 0 or x = 3

Example 2

When are the values of the function f negative?

Example 3

At what values of the variable x does the function become positive and at what values of x does it become negative?

Solution

The zeros of the numerator are x = 0 and x = 4. The zeros of the denominator are x = -1 and x = 3. The function is not defined at the denominator zeros.

The graphs of both the numerator and the denominator are upward-opening parabolas, so they are negative between their zeros. Let's make a table of signs.

Since the function is not defined in the zeros of the denominator, it is plotted in the character diagram by crossing these points over the quotient.

According to the sign rules of the division calculation, the signs are obtained for the quotient. The answer can be read from the table of signs. That is, the function f (x) gets positive values when -1 < x, 0 < x < 3 and 4 < x, and negative values when -1 < x < 0 and 3 < x < 4. Below is a graph of the function and graphs of the numerator and denominator.

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