Skill: Determine intervals where a function is positive or negative

To find where a function is positive or negative:

• Positive: $f(x)>0$, so the graph is above the $x$-axis.
• Negative: $f(x)<0$, so the graph is below the $x$-axis.
• Zeros and undefined inputs cannot be included in positive or negative intervals.

Example

Determine where

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

is positive and negative.

Step 1: Find zeros

A function is zero when its numerator is zero:

$$(x-2)^2=0$$

So,

$$x=2$$

The value $x=2$ is a zero, so it cannot be included in a positive or negative interval.

Step 2: Find values excluded from the domain

The denominator cannot equal zero:

$$x+1=0$$

Thus,

$$x=-1$$

The function is undefined at $x=-1$, so this value also cannot be included.

These critical values divide the number line into three intervals:

$$(-\infty,-1), \qquad (-1,2), \qquad (2,\infty)$$

Step 3: Test the sign in each interval

Choose one test value from each interval.

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

For example,

$$f(-2)=\frac{(-2-2)^2}{-2+1} =\frac{16}{-1}<0$$

and

$$f(0)=\frac{(0-2)^2}{0+1} =\frac{4}{1}>0$$

The value $f(3)$ is also positive.

Step 4: Write the intervals

The function is negative on

$$\boxed{(-\infty,-1)}$$

The function is positive on

$$\boxed{(-1,2)\cup(2,\infty)}$$

Use parentheses because $x=-1$ is undefined and $x=2$ makes the function equal to zero. Notice that the function is positive on both sides of $x=2$, but the intervals must still be separated because $f(2)=0$.