Determine intervals where a function is positive or negative

Positive and negative intervals are the subsets of a function’s domain on which $f(x)>0$ or $f(x)<0$ , represented graphically by portions above or below the $x$ -axis and algebraically through zeros, domain restrictions, and sign analysis. Correct interval notation excludes zeros and undefined inputs, and a zero does not necessarily mark a sign change. The focus is on polynomial and rational functions in familiar forms, rather than advanced generalized or parameterized sign analysis.

Detailed Explanation: 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 $f(2)=0$ .

Learn by doing: Determine intervals where a function is positive or negative

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Polynomial Inequalities - Expanded Quadratic - Intervals

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