Determine intervals where a function is positive or negative

Determining where a function is positive or negative means identifying the intervals in its real domain where $f(x)>0$ or $f(x)<0$ , using its graph, table, or algebraic form. The analysis relates sign changes to x-intercepts, zeros, and domain restrictions such as vertical asymptotes, while recognizing that a function may touch the x-axis without changing sign; boundary zeros and excluded values are not included in the intervals. Complex-valued functions and abstract domains are outside this scope.

Detailed Explanation: 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.

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

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Rational Function Inequalities - Factored Quadratic over Binomial - Sign Chart

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