Calculus Examples

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

Step 1

Set the radicand in $\sqrt{x^{2} + 9}$ greater than or equal to $0$ to find where the expression is defined.

$x^{2} + 9 \geq 0$

Step 2

Solve for $x$ .

Tap for more steps...

Step 2.1

Subtract $9$ from both sides of the inequality.

$x^{2} \geq - 9$

Step 2.2

Since the left side has an even power, it is always positive for all real numbers.

All real numbers

Step 3

Set the denominator in $\frac{5 (\sqrt{x^{2} + 9})}{x}$ 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:

$(- \infty, - 5) \cup (5, \infty)$

Set-Builder Notation:

${y | y < - 5, y > 5}$

Step 6

Determine the domain and range.

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

Range: $(- \infty, - 5) \cup (5, \infty), {y | y < - 5, y > 5}$

Step 7