Skill: Determine intervals where a function is positive or negative

To determine where a function is positive or negative, find the important $x$-values where its sign could change:

• Zeros: values where $f(x)=0$
• Excluded values: values where the function is undefined, such as a vertical asymptote

These values divide the number line into intervals. Test one number from each interval to determine whether $f(x)$ is positive or negative.

Example

Determine where

$$f(x)=\frac{(x-2)(x+1)}{x-3}$$

is positive and negative.

1. Find the zeros and excluded values

The numerator is zero when

$$x-2=0 \quad\text{or}\quad x+1=0,$$

so the zeros are

$$x=2 \quad\text{and}\quad x=-1.$$

The denominator is zero when

$$x-3=0,$$

so $x=3$ is excluded from the domain.

These values divide the number line into four intervals:

$$(-\infty,-1),\quad (-1,2),\quad (2,3),\quad (3,\infty).$$

2. Test one value in each interval

Interval	Test value	Sign of $f(x)$
$(-\infty,-1)$	$x=-2$	Negative
$(-1,2)$	$x=0$	Positive
$(2,3)$	$x=2.5$	Negative
$(3,\infty)$	$x=4$	Positive

For example, at $x=0$,

$$f(0)=\frac{(0-2)(0+1)}{0-3} =\frac{(-)(+)}{(-)}>0.$$

Thus, $f(x)$ is positive on the interval containing $0$.

3. State the answer

The function is positive where

$$\boxed{f(x)>0\text{ on }(-1,2)\cup(3,\infty)}$$

The function is negative where

$$\boxed{f(x)<0\text{ on }(-\infty,-1)\cup(2,3)}$$

The endpoints are not included because $f(-1)=0$ and $f(2)=0$, while $f(3)$ is undefined.