Algebra Examples

$f (x) = \frac{1}{9} x^{2} + 2$ , $x \geq 0$

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.

$[2, \infty)$

Step 1.2

Convert $[2, \infty)$ to an inequality.

$y \geq 2$

Step 2

Find the inverse.

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Step 2.1

Interchange the variables.

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

Step 2.2

Solve for $y$ .

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Step 2.2.1

Rewrite the equation as $\frac{1}{9} y^{2} + 2 = x$ .

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

Step 2.2.2

Combine $\frac{1}{9}$ and $y^{2}$ .

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

Step 2.2.3

Subtract $2$ from both sides of the equation.

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

Step 2.2.4

Multiply both sides of the equation by $9$ .

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

Step 2.2.5

Simplify both sides of the equation.

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Step 2.2.5.1

Simplify the left side.

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Step 2.2.5.1.1

Cancel the common factor of $9$ .

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Step 2.2.5.1.1.1

Cancel the common factor.

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

Step 2.2.5.1.1.2

Rewrite the expression.

$y^{2} = 9 (x - 2)$

Step 2.2.5.2

Simplify the right side.

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Step 2.2.5.2.1

Simplify $9 (x - 2)$ .

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Step 2.2.5.2.1.1

Apply the distributive property.

$y^{2} = 9 x + 9 \cdot - 2$

Step 2.2.5.2.1.2

Multiply $9$ by $- 2$ .

$y^{2} = 9 x - 18$

Step 2.2.6

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

$y = \pm \sqrt{9 x - 18}$

Step 2.2.7

Simplify $\pm \sqrt{9 x - 18}$ .

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Step 2.2.7.1

Factor $9$ out of $9 x - 18$ .

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Step 2.2.7.1.1

Factor $9$ out of $9 x$ .

$y = \pm \sqrt{9 (x) - 18}$

Step 2.2.7.1.2

Factor $9$ out of $- 18$ .

$y = \pm \sqrt{9 x + 9 \cdot - 2}$

Step 2.2.7.1.3

Factor $9$ out of $9 x + 9 \cdot - 2$ .

$y = \pm \sqrt{9 (x - 2)}$

Step 2.2.7.2

Rewrite $9$ as $3^{2}$ .

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

Step 2.2.7.3

Pull terms out from under the radical.

$y = \pm 3 \sqrt{x - 2}$

Step 2.2.8

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

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Step 2.2.8.1

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

$y = 3 \sqrt{x - 2}$

Step 2.2.8.2

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

$y = - 3 \sqrt{x - 2}$

Step 2.2.8.3

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

$y = 3 \sqrt{x - 2}$

$y = - 3 \sqrt{x - 2}$

$y = 3 \sqrt{x - 2}$

$y = - 3 \sqrt{x - 2}$

$y = 3 \sqrt{x - 2}$

$y = - 3 \sqrt{x - 2}$

Step 2.3

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

$f^{- 1} (x) = 3 \sqrt{x - 2}, - 3 \sqrt{x - 2}$

Step 3

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

$f^{- 1} (x) = 3 \sqrt{x - 2}, x \geq 2$

Step 4