Precalculus Examples

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

Step 1

Write $f (x) = \frac{\sqrt{x}}{2}$ as an equation.

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{\sqrt{y}}{2} = x$ .

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

Step 3.2

Multiply both sides of the equation by $2$ .

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

Step 3.3

Simplify the left side.

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Step 3.3.1

Cancel the common factor of $2$ .

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Step 3.3.1.1

Cancel the common factor.

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

Step 3.3.1.2

Rewrite the expression.

$\sqrt{y} = 2 x$

Step 3.4

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

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

Step 3.5

Simplify each side of the equation.

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Step 3.5.1

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

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

Step 3.5.2

Simplify the left side.

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Step 3.5.2.1

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

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Step 3.5.2.1.1

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

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Step 3.5.2.1.1.1

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

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

Step 3.5.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.5.2.1.1.2.1

Cancel the common factor.

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

Step 3.5.2.1.1.2.2

Rewrite the expression.

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

Step 3.5.2.1.2

Simplify.

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

Step 3.5.3

Simplify the right side.

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Step 3.5.3.1

Simplify ${(2 x)}^{2}$ .

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Step 3.5.3.1.1

Apply the product rule to $2 x$ .

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

Step 3.5.3.1.2

Raise $2$ to the power of $2$ .

$y = 4 x^{2}$

Step 4

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

$f^{- 1} (x) = 4 x^{2}$

Step 5

Verify if $f^{- 1} (x) = 4 x^{2}$ is the inverse of $f (x) = \frac{\sqrt{x}}{2}$ .

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Step 5.1

To verify the inverse, check if $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ .

Step 5.2

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

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Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

Evaluate $f^{- 1} (\frac{\sqrt{x}}{2})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{\sqrt{x}}{2}) = 4 {(\frac{\sqrt{x}}{2})}^{2}$

Step 5.2.3

Apply the product rule to $\frac{\sqrt{x}}{2}$ .

$f^{- 1} (\frac{\sqrt{x}}{2}) = 4 (\frac{{\sqrt{x}}^{2}}{2^{2}})$

Step 5.2.4

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

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Step 5.2.4.1

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

$f^{- 1} (\frac{\sqrt{x}}{2}) = 4 (\frac{{(x^{\frac{1}{2}})}^{2}}{2^{2}})$

Step 5.2.4.2

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

$f^{- 1} (\frac{\sqrt{x}}{2}) = 4 (\frac{x^{\frac{1}{2} \cdot 2}}{2^{2}})$

Step 5.2.4.3

Combine $\frac{1}{2}$ and $2$ .

$f^{- 1} (\frac{\sqrt{x}}{2}) = 4 (\frac{x^{\frac{2}{2}}}{2^{2}})$

Step 5.2.4.4

Cancel the common factor of $2$ .

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Step 5.2.4.4.1

Cancel the common factor.

$f^{- 1} (\frac{\sqrt{x}}{2}) = 4 (\frac{x^{\frac{2}{2}}}{2^{2}})$

Step 5.2.4.4.2

Rewrite the expression.

$f^{- 1} (\frac{\sqrt{x}}{2}) = 4 (\frac{x}{2^{2}})$

Step 5.2.4.5

Simplify.

$f^{- 1} (\frac{\sqrt{x}}{2}) = 4 (\frac{x}{2^{2}})$

Step 5.2.5

Raise $2$ to the power of $2$ .

$f^{- 1} (\frac{\sqrt{x}}{2}) = 4 (\frac{x}{4})$

Step 5.2.6

Cancel the common factor of $4$ .

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Step 5.2.6.1

Cancel the common factor.

$f^{- 1} (\frac{\sqrt{x}}{2}) = 4 (\frac{x}{4})$

Step 5.2.6.2

Rewrite the expression.

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

Step 5.3

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

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Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

Evaluate $f (4 x^{2})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (4 x^{2}) = \frac{\sqrt{4 x^{2}}}{2}$

Step 5.3.3

Simplify the numerator.

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Step 5.3.3.1

Rewrite $4 x^{2}$ as ${(2 x)}^{2}$ .

$f (4 x^{2}) = \frac{\sqrt{{(2 x)}^{2}}}{2}$

Step 5.3.3.2

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

$f (4 x^{2}) = \frac{2 x}{2}$

Step 5.3.4

Cancel the common factor of $2$ .

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Step 5.3.4.1

Cancel the common factor.

$f (4 x^{2}) = \frac{2 x}{2}$

Step 5.3.4.2

Divide $x$ by $1$ .

$f (4 x^{2}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 4 x^{2}$ is the inverse of $f (x) = \frac{\sqrt{x}}{2}$ .

$f^{- 1} (x) = 4 x^{2}$