Examples

f (x) = \sqrt{x}

$f (x) = \sqrt{x}$

Step 1

Write

f (x) = \sqrt{x}

$f (x) = \sqrt{x}$ as an equation.

y = \sqrt{x}

$y = \sqrt{x}$

Step 2

Interchange the variables.

x = \sqrt{y}

$x = \sqrt{y}$

Step 3

Solve for

y

$y$ .

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Step 3.1

Rewrite the equation as

\sqrt{y} = x

$\sqrt{y} = x$ .

\sqrt{y} = x

$\sqrt{y} = x$

Step 3.2

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

{\sqrt{y}}^{2} = x^{2}

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

Step 3.3

Simplify each side of the equation.

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Step 3.3.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{y}

$\sqrt{y}$ as

y^{\frac{1}{2}}

$y^{\frac{1}{2}}$ .

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

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

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Simplify

{(y^{\frac{1}{2}})}^{2}

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

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Step 3.3.2.1.1

Multiply the exponents in

{(y^{\frac{1}{2}})}^{2}

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

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Step 3.3.2.1.1.1

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

y^{\frac{1}{2} \cdot 2} = x^{2}

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

Step 3.3.2.1.1.2

Cancel the common factor of

2

$2$ .

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Step 3.3.2.1.1.2.1

Cancel the common factor.

y^{\frac{1}{2} \cdot 2} = x^{2}

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

Step 3.3.2.1.1.2.2

Rewrite the expression.

y^{1} = x^{2}

$y^{1} = x^{2}$

y^{1} = x^{2}

$y^{1} = x^{2}$

y^{1} = x^{2}

$y^{1} = x^{2}$

Step 3.3.2.1.2

Simplify.

y = x^{2}

$y = x^{2}$

y = x^{2}

$y = x^{2}$

y = x^{2}

$y = x^{2}$

y = x^{2}

$y = x^{2}$

y = x^{2}

$y = x^{2}$

Step 4

Replace

y

$y$ with

f^{- 1} (x)

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

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

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

Step 5

Verify if

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

$f^{- 1} (x) = x^{2}$ is the inverse of

f (x) = \sqrt{x}

$f (x) = \sqrt{x}$ .

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Step 5.1

To verify the inverse, check if

f^{- 1} (f (x)) = x

$f^{- 1} (f (x)) = x$ and

f (f^{- 1} (x)) = x

$f (f^{- 1} (x)) = x$ .

Step 5.2

Evaluate

f^{- 1} (f (x))

$f^{- 1} (f (x))$ .

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Step 5.2.1

Set up the composite result function.

f^{- 1} (f (x))

$f^{- 1} (f (x))$

Step 5.2.2

Evaluate

f^{- 1} (\sqrt{x})

$f^{- 1} (\sqrt{x})$ by substituting in the value of

f

$f$ into

f^{- 1}

$f^{- 1}$ .

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

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

Step 5.2.3

Rewrite

{\sqrt{x}}^{2}

${\sqrt{x}}^{2}$ as

x

$x$ .

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Step 5.2.3.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{x}

$\sqrt{x}$ as

x^{\frac{1}{2}}

$x^{\frac{1}{2}}$ .

f^{- 1} (\sqrt{x}) = {(x^{\frac{1}{2}})}^{2}

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

Step 5.2.3.2

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

f^{- 1} (\sqrt{x}) = x^{\frac{1}{2} \cdot 2}

$f^{- 1} (\sqrt{x}) = x^{\frac{1}{2} \cdot 2}$

Step 5.2.3.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

f^{- 1} (\sqrt{x}) = x^{\frac{2}{2}}

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

Step 5.2.3.4

Cancel the common factor of

2

$2$ .

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Step 5.2.3.4.1

Cancel the common factor.

f^{- 1} (\sqrt{x}) = x^{\frac{2}{2}}

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

Step 5.2.3.4.2

Rewrite the expression.

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

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

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

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

Step 5.2.3.5

Simplify.

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

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

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

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

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

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

Step 5.3

Evaluate

f (f^{- 1} (x))

$f (f^{- 1} (x))$ .

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Step 5.3.1

Set up the composite result function.

f (f^{- 1} (x))

$f (f^{- 1} (x))$

Step 5.3.2

Evaluate

f (x^{2})

$f (x^{2})$ by substituting in the value of

f^{- 1}

$f^{- 1}$ into

f

$f$ .

f (x^{2}) = \sqrt{x^{2}}

$f (x^{2}) = \sqrt{x^{2}}$

Step 5.3.3

Remove parentheses.

f (x^{2}) = \sqrt{x^{2}}

$f (x^{2}) = \sqrt{x^{2}}$

Step 5.3.4

Pull terms out from under the radical, assuming positive real numbers.

f (x^{2}) = x

$f (x^{2}) = x$

f (x^{2}) = x

$f (x^{2}) = x$

Step 5.4

Since

f^{- 1} (f (x)) = x

$f^{- 1} (f (x)) = x$ and

f (f^{- 1} (x)) = x

$f (f^{- 1} (x)) = x$ , then

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

$f^{- 1} (x) = x^{2}$ is the inverse of

f (x) = \sqrt{x}

$f (x) = \sqrt{x}$ .

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

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

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

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

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