Sign of a rational function
A rational function is
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.
Numerator
Binomial square
Denominator
Zero-product property
The zero point of the function f is x = 2 and the function is not defined when x = 0 or x = 3
In examining the sign of a rational function, the signs of the numerator and the signs of the denominator are examined separately. These are used to find the signs of the rational function itself. Just remember these simple calculation rules.
If the dividend and the divider are of the same sign, the result is positive. If the dividend and the divider are of different signs, the result is negative.
Example 2
When are the values of the function f negative?
The function is not defined when x = 0. The denominator of a function is positive for all values of x, so the sign is determined only by the numerator.
We can see from the graph that f(x) < 0, when x < -2
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.
Numerator
Negative values, when 0 < x < 4
Denominator
Negative values, when -1 < x < 3
Function f(x)
Negative values, when -1 < x < 0 and 3 < x < 4
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