Algebra Examples

$f (x) = x^{2} + 5$ , $x \geq 0$

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.

$[5, \infty)$

Step 1.2

Convert $[5, \infty)$ to an inequality.

$y \geq 5$

Step 2

Find the inverse.

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Step 2.1

Interchange the variables.

$x = y^{2} + 5$

Step 2.2

Solve for $y$ .

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Step 2.2.1

Rewrite the equation as $y^{2} + 5 = x$ .

$y^{2} + 5 = x$

Step 2.2.2

Subtract $5$ from both sides of the equation.

$y^{2} = x - 5$

Step 2.2.3

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

$y = \pm \sqrt{x - 5}$

Step 2.2.4

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

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Step 2.2.4.1

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

$y = \sqrt{x - 5}$

Step 2.2.4.2

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

$y = - \sqrt{x - 5}$

Step 2.2.4.3

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

$y = \sqrt{x - 5}$

$y = - \sqrt{x - 5}$

$y = \sqrt{x - 5}$

$y = - \sqrt{x - 5}$

$y = \sqrt{x - 5}$

$y = - \sqrt{x - 5}$

Step 2.3

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

$f^{- 1} (x) = \sqrt{x - 5}, - \sqrt{x - 5}$

Step 3

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

$f^{- 1} (x) = \sqrt{x - 5}, x \geq 5$

Step 4