Algebra Examples

$f (x) = 9 x^{2}$ , $x \leq 0$

Step 1

Find the range of the given function.

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Step 1.1

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

$[0, \infty)$

Step 1.2

Convert $[0, \infty)$ to an inequality.

$y \geq 0$

Step 2

Find the inverse.

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Step 2.1

Interchange the variables.

$x = 9 y^{2}$

Step 2.2

Solve for $y$ .

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Step 2.2.1

Rewrite the equation as $9 y^{2} = x$ .

$9 y^{2} = x$

Step 2.2.2

Divide each term in $9 y^{2} = x$ by $9$ and simplify.

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Step 2.2.2.1

Divide each term in $9 y^{2} = x$ by $9$ .

$\frac{9 y^{2}}{9} = \frac{x}{9}$

Step 2.2.2.2

Simplify the left side.

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Step 2.2.2.2.1

Cancel the common factor of $9$ .

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Step 2.2.2.2.1.1

Cancel the common factor.

$\frac{9 y^{2}}{9} = \frac{x}{9}$

Step 2.2.2.2.1.2

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

$y^{2} = \frac{x}{9}$

Step 2.2.3

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

$y = \pm \sqrt{\frac{x}{9}}$

Step 2.2.4

Simplify $\pm \sqrt{\frac{x}{9}}$ .

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Step 2.2.4.1

Rewrite $\frac{x}{9}$ as ${(\frac{1}{3})}^{2} x$ .

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Step 2.2.4.1.1

Factor the perfect power $1^{2}$ out of $x$ .

$y = \pm \sqrt{\frac{1^{2} x}{9}}$

Step 2.2.4.1.2

Factor the perfect power $3^{2}$ out of $9$ .

$y = \pm \sqrt{\frac{1^{2} x}{3^{2} \cdot 1}}$

Step 2.2.4.1.3

Rearrange the fraction $\frac{1^{2} x}{3^{2} \cdot 1}$ .

$y = \pm \sqrt{{(\frac{1}{3})}^{2} x}$

Step 2.2.4.2

Pull terms out from under the radical.

$y = \pm \frac{1}{3} \sqrt{x}$

Step 2.2.4.3

Combine $\frac{1}{3}$ and $\sqrt{x}$ .

$y = \pm \frac{\sqrt{x}}{3}$

Step 2.2.5

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

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Step 2.2.5.1

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

$y = \frac{\sqrt{x}}{3}$

Step 2.2.5.2

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

$y = - \frac{\sqrt{x}}{3}$

Step 2.2.5.3

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

$y = \frac{\sqrt{x}}{3}$

$y = - \frac{\sqrt{x}}{3}$

$y = \frac{\sqrt{x}}{3}$

$y = - \frac{\sqrt{x}}{3}$

$y = \frac{\sqrt{x}}{3}$

$y = - \frac{\sqrt{x}}{3}$

Step 2.3

Replace $y$ with $f^{- 1} (x)$ to show the final answer.

$f^{- 1} (x) = \frac{\sqrt{x}}{3}, - \frac{\sqrt{x}}{3}$

Step 3

Find the inverse using the domain and the range of the original function.

$f^{- 1} (x) = - \frac{\sqrt{x}}{3}, x \geq 0$

Step 4