Calculus Examples

$f (x) = \frac{5 x}{x^{2 - 9}}$

Step 1

Set the denominator in $\frac{5 x}{x^{2 - 9}}$ equal to $0$ to find where the expression is undefined.

$x^{2 - 9} = 0$

Step 2

Solve for $x$ .

Tap for more steps...

Step 2.1

Simplify $x^{2 - 9}$ .

Tap for more steps...

Step 2.1.1

Subtract $9$ from $2$ .

$x^{- 7} = 0$

Step 2.1.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{x^{7}} = 0$

Step 2.2

Set the numerator equal to zero.

$1 = 0$

Step 2.3

Since $1 \neq 0$ , there are no solutions.

No solution

Step 3

Set the base in $x^{2 - 9}$ equal to $0$ to find where the expression is undefined.

$x = 0$

Step 4

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, 0) \cup (0, \infty)$

Set-Builder Notation:

${x | x \neq 0}$

Step 5

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${y | y \geq 0}$

Step 6

Determine the domain and range.

Domain: $(- \infty, 0) \cup (0, \infty), {x | x \neq 0}$

Range: $[0, \infty), {y | y \geq 0}$

Step 7