Precalculus Examples

$f (x) = \frac{x^{2} - 9}{2 x^{2} + 1}$

Step 1

Set the denominator in $\frac{x^{2} - 9}{2 x^{2} + 1}$ equal to $0$ to find where the expression is undefined.

$2 x^{2} + 1 = 0$

Step 2

Solve for $x$ .

Tap for more steps...

Step 2.1

Subtract $1$ from both sides of the equation.

$2 x^{2} = - 1$

Step 2.2

Divide each term in $2 x^{2} = - 1$ by $2$ and simplify.

Tap for more steps...

Step 2.2.1

Divide each term in $2 x^{2} = - 1$ by $2$ .

$\frac{2 x^{2}}{2} = \frac{- 1}{2}$

Step 2.2.2

Simplify the left side.

Tap for more steps...

Step 2.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.2.2.1.1

Cancel the common factor.

$\frac{2 x^{2}}{2} = \frac{- 1}{2}$

Step 2.2.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{- 1}{2}$

Step 2.2.3

Simplify the right side.

Tap for more steps...

Step 2.2.3.1

Move the negative in front of the fraction.

$x^{2} = - \frac{1}{2}$

Step 2.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{- \frac{1}{2}}$

Step 2.4

Simplify $\pm \sqrt{- \frac{1}{2}}$ .

Tap for more steps...

Step 2.4.1

Rewrite $- \frac{1}{2}$ as $i^{2} \frac{1^{2}}{2}$ .

Tap for more steps...

Step 2.4.1.1

Rewrite $- 1$ as $i^{2}$ .

$x = \pm \sqrt{i^{2} \frac{1}{2}}$

Step 2.4.1.2

Rewrite $1$ as $1^{2}$ .

$x = \pm \sqrt{i^{2} \frac{1^{2}}{2}}$

Step 2.4.2

Pull terms out from under the radical.

$x = \pm i \sqrt{\frac{1^{2}}{2}}$

Step 2.4.3

One to any power is one.

$x = \pm i \sqrt{\frac{1}{2}}$

Step 2.4.4

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

$x = \pm i \frac{\sqrt{1}}{\sqrt{2}}$

Step 2.4.5

Any root of $1$ is $1$ .

$x = \pm i \frac{1}{\sqrt{2}}$

Step 2.4.6

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \pm i (\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})$

Step 2.4.7

Combine and simplify the denominator.

Tap for more steps...

Step 2.4.7.1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \pm i \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 2.4.7.2

Raise $\sqrt{2}$ to the power of $1$ .

$x = \pm i \frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 2.4.7.3

Raise $\sqrt{2}$ to the power of $1$ .

$x = \pm i \frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 2.4.7.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \pm i \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 2.4.7.5

Add $1$ and $1$ .

$x = \pm i \frac{\sqrt{2}}{{\sqrt{2}}^{2}}$

Step 2.4.7.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Tap for more steps...

Step 2.4.7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$x = \pm i \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 2.4.7.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \pm i \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 2.4.7.6.3

Combine $\frac{1}{2}$ and $2$ .

$x = \pm i \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 2.4.7.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.4.7.6.4.1

Cancel the common factor.

$x = \pm i \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 2.4.7.6.4.2

Rewrite the expression.

$x = \pm i \frac{\sqrt{2}}{2^{1}}$

Step 2.4.7.6.5

Evaluate the exponent.

$x = \pm i \frac{\sqrt{2}}{2}$

Step 2.4.8

Combine $i$ and $\frac{\sqrt{2}}{2}$ .

$x = \pm \frac{i \sqrt{2}}{2}$

Step 2.5

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps...

Step 2.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{i \sqrt{2}}{2}$

Step 2.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{i \sqrt{2}}{2}$

Step 2.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{i \sqrt{2}}{2}, - \frac{i \sqrt{2}}{2}$

Step 3

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 4

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

Interval Notation:

$[- 9, \frac{1}{2})$

Set-Builder Notation:

${y | - 9 \leq y < \frac{1}{2}}$

Step 5

Determine the domain and range.

Domain: $(- \infty, \infty), {x | x \in ℝ}$

Range: $[- 9, \frac{1}{2}), {y | - 9 \leq y < \frac{1}{2}}$

Step 6