Algebra Examples

$f (x) = \sqrt{x + 9}$ , $x \geq - 9$

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.

$x = \sqrt{y + 9}$

Step 2.2

Solve for $y$ .

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Step 2.2.1

Rewrite the equation as $\sqrt{y + 9} = x$ .

$\sqrt{y + 9} = x$

Step 2.2.2

To remove the radical on the left side of the equation, square both sides of the equation.

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

Step 2.2.3

Simplify each side of the equation.

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Step 2.2.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y + 9}$ as ${(y + 9)}^{\frac{1}{2}}$ .

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

Step 2.2.3.2

Simplify the left side.

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Step 2.2.3.2.1

Simplify ${({(y + 9)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.3.2.1.1

Multiply the exponents in ${({(y + 9)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.3.2.1.1.1

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

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

Step 2.2.3.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.3.2.1.1.2.1

Cancel the common factor.

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

Step 2.2.3.2.1.1.2.2

Rewrite the expression.

${(y + 9)}^{1} = x^{2}$

Step 2.2.3.2.1.2

Simplify.

$y + 9 = x^{2}$

Step 2.2.4

Subtract $9$ from both sides of the equation.

$y = x^{2} - 9$

Step 2.3

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

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

Step 3

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

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

Step 4