Precalculus Examples

$f (x) = \sqrt{4 x + 12}$

Step 1

Write $f (x) = \sqrt{4 x + 12}$ as an equation.

$y = \sqrt{4 x + 12}$

Step 2

Interchange the variables.

$x = \sqrt{4 y + 12}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\sqrt{4 y + 12} = x$ .

$\sqrt{4 y + 12} = x$

Step 3.2

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

${\sqrt{4 y + 12}}^{2} = x^{2}$

Step 3.3

Simplify each side of the equation.

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Step 3.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 y + 12}$ as ${(4 y + 12)}^{\frac{1}{2}}$ .

${({(4 y + 12)}^{\frac{1}{2}})}^{2} = x^{2}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Simplify ${({(4 y + 12)}^{\frac{1}{2}})}^{2}$ .

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Step 3.3.2.1.1

Multiply the exponents in ${({(4 y + 12)}^{\frac{1}{2}})}^{2}$ .

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Step 3.3.2.1.1.1

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

${(4 y + 12)}^{\frac{1}{2} \cdot 2} = x^{2}$

Step 3.3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.3.2.1.1.2.1

Cancel the common factor.

${(4 y + 12)}^{\frac{1}{2} \cdot 2} = x^{2}$

Step 3.3.2.1.1.2.2

Rewrite the expression.

${(4 y + 12)}^{1} = x^{2}$

Step 3.3.2.1.2

Simplify.

$4 y + 12 = x^{2}$

Step 3.4

Solve for $y$ .

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Step 3.4.1

Subtract $12$ from both sides of the equation.

$4 y = x^{2} - 12$

Step 3.4.2

Divide each term in $4 y = x^{2} - 12$ by $4$ and simplify.

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Step 3.4.2.1

Divide each term in $4 y = x^{2} - 12$ by $4$ .

$\frac{4 y}{4} = \frac{x^{2}}{4} + \frac{- 12}{4}$

Step 3.4.2.2

Simplify the left side.

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Step 3.4.2.2.1

Cancel the common factor of $4$ .

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Step 3.4.2.2.1.1

Cancel the common factor.

$\frac{4 y}{4} = \frac{x^{2}}{4} + \frac{- 12}{4}$

Step 3.4.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{x^{2}}{4} + \frac{- 12}{4}$

Step 3.4.2.3

Simplify the right side.

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Step 3.4.2.3.1

Divide $- 12$ by $4$ .

$y = \frac{x^{2}}{4} - 3$

Step 4

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

$f^{- 1} (x) = \frac{x^{2}}{4} - 3$

Step 5

Verify if $f^{- 1} (x) = \frac{x^{2}}{4} - 3$ is the inverse of $f (x) = \sqrt{4 x + 12}$ .

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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 $f^{- 1} (\sqrt{4 x + 12})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\sqrt{4 x + 12}) = \frac{{(\sqrt{4 x + 12})}^{2}}{4} - 3$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Simplify the numerator.

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Step 5.2.3.1.1

Factor $4$ out of $4 x + 12$ .

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Step 5.2.3.1.1.1

Factor $4$ out of $4 x$ .

$f^{- 1} (\sqrt{4 x + 12}) = \frac{{\sqrt{4 (x) + 12}}^{2}}{4} - 3$

Step 5.2.3.1.1.2

Factor $4$ out of $12$ .

$f^{- 1} (\sqrt{4 x + 12}) = \frac{{\sqrt{4 x + 4 \cdot 3}}^{2}}{4} - 3$

Step 5.2.3.1.1.3

Factor $4$ out of $4 x + 4 \cdot 3$ .

$f^{- 1} (\sqrt{4 x + 12}) = \frac{{\sqrt{4 (x + 3)}}^{2}}{4} - 3$

Step 5.2.3.1.2

Rewrite $4$ as $2^{2}$ .

$f^{- 1} (\sqrt{4 x + 12}) = \frac{{\sqrt{2^{2} (x + 3)}}^{2}}{4} - 3$

Step 5.2.3.1.3

Pull terms out from under the radical.

$f^{- 1} (\sqrt{4 x + 12}) = \frac{{(2 \sqrt{x + 3})}^{2}}{4} - 3$

Step 5.2.3.1.4

Apply the product rule to $2 \sqrt{x + 3}$ .

$f^{- 1} (\sqrt{4 x + 12}) = \frac{2^{2} {\sqrt{x + 3}}^{2}}{4} - 3$

Step 5.2.3.1.5

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

$f^{- 1} (\sqrt{4 x + 12}) = \frac{4 {\sqrt{x + 3}}^{2}}{4} - 3$

Step 5.2.3.1.6

Rewrite ${\sqrt{x + 3}}^{2}$ as $x + 3$ .

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Step 5.2.3.1.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 3}$ as ${(x + 3)}^{\frac{1}{2}}$ .

$f^{- 1} (\sqrt{4 x + 12}) = \frac{4 {({(x + 3)}^{\frac{1}{2}})}^{2}}{4} - 3$

Step 5.2.3.1.6.2

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

$f^{- 1} (\sqrt{4 x + 12}) = \frac{4 {(x + 3)}^{\frac{1}{2} \cdot 2}}{4} - 3$

Step 5.2.3.1.6.3

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

$f^{- 1} (\sqrt{4 x + 12}) = \frac{4 {(x + 3)}^{\frac{2}{2}}}{4} - 3$

Step 5.2.3.1.6.4

Cancel the common factor of $2$ .

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Step 5.2.3.1.6.4.1

Cancel the common factor.

$f^{- 1} (\sqrt{4 x + 12}) = \frac{4 {(x + 3)}^{\frac{2}{2}}}{4} - 3$

Step 5.2.3.1.6.4.2

Rewrite the expression.

$f^{- 1} (\sqrt{4 x + 12}) = \frac{4 (x + 3)}{4} - 3$

Step 5.2.3.1.6.5

Simplify.

$f^{- 1} (\sqrt{4 x + 12}) = \frac{4 (x + 3)}{4} - 3$

Step 5.2.3.1.7

Apply the distributive property.

$f^{- 1} (\sqrt{4 x + 12}) = \frac{4 x + 4 \cdot 3}{4} - 3$

Step 5.2.3.1.8

Multiply $4$ by $3$ .

$f^{- 1} (\sqrt{4 x + 12}) = \frac{4 x + 12}{4} - 3$

Step 5.2.3.1.9

Factor $4$ out of $4 x + 12$ .

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Step 5.2.3.1.9.1

Factor $4$ out of $4 x$ .

$f^{- 1} (\sqrt{4 x + 12}) = \frac{4 (x) + 12}{4} - 3$

Step 5.2.3.1.9.2

Factor $4$ out of $12$ .

$f^{- 1} (\sqrt{4 x + 12}) = \frac{4 x + 4 \cdot 3}{4} - 3$

Step 5.2.3.1.9.3

Factor $4$ out of $4 x + 4 \cdot 3$ .

$f^{- 1} (\sqrt{4 x + 12}) = \frac{4 (x + 3)}{4} - 3$

Step 5.2.3.2

Cancel the common factor of $4$ .

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Step 5.2.3.2.1

Cancel the common factor.

$f^{- 1} (\sqrt{4 x + 12}) = \frac{4 (x + 3)}{4} - 3$

Step 5.2.3.2.2

Divide $x + 3$ by $1$ .

$f^{- 1} (\sqrt{4 x + 12}) = x + 3 - 3$

Step 5.2.4

Combine the opposite terms in $x + 3 - 3$ .

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Step 5.2.4.1

Subtract $3$ from $3$ .

$f^{- 1} (\sqrt{4 x + 12}) = x + 0$

Step 5.2.4.2

Add $x$ and $0$ .

$f^{- 1} (\sqrt{4 x + 12}) = x$

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 $f (\frac{x^{2}}{4} - 3)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x^{2}}{4} - 3) = \sqrt{4 (\frac{x^{2}}{4} - 3) + 12}$

Step 5.3.3

Factor $4$ out of $4 (\frac{x^{2}}{4} - 3) + 12$ .

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Step 5.3.3.1

Factor $4$ out of $12$ .

$f (\frac{x^{2}}{4} - 3) = \sqrt{4 (\frac{x^{2}}{4} - 3) + 4 \cdot 3}$

Step 5.3.3.2

Factor $4$ out of $4 (\frac{x^{2}}{4} - 3) + 4 \cdot 3$ .

$f (\frac{x^{2}}{4} - 3) = \sqrt{4 (\frac{x^{2}}{4} - 3 + 3)}$

Step 5.3.4

Simplify by adding numbers.

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Step 5.3.4.1

Add $- 3$ and $3$ .

$f (\frac{x^{2}}{4} - 3) = \sqrt{4 (\frac{x^{2}}{4} + 0)}$

Step 5.3.4.2

Add $\frac{x^{2}}{4}$ and $0$ .

$f (\frac{x^{2}}{4} - 3) = \sqrt{4 (\frac{x^{2}}{4})}$

Step 5.3.5

Combine $4$ and $\frac{x^{2}}{4}$ .

$f (\frac{x^{2}}{4} - 3) = \sqrt{\frac{4 x^{2}}{4}}$

Step 5.3.6

Reduce the expression by cancelling the common factors.

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Step 5.3.6.1

Reduce the expression $\frac{4 x^{2}}{4}$ by cancelling the common factors.

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Step 5.3.6.1.1

Cancel the common factor.

$f (\frac{x^{2}}{4} - 3) = \sqrt{\frac{4 x^{2}}{4}}$

Step 5.3.6.1.2

Rewrite the expression.

$f (\frac{x^{2}}{4} - 3) = \sqrt{\frac{x^{2}}{1}}$

Step 5.3.6.2

Divide $x^{2}$ by $1$ .

$f (\frac{x^{2}}{4} - 3) = \sqrt{x^{2}}$

Step 5.3.7

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

$f (\frac{x^{2}}{4} - 3) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x^{2}}{4} - 3$ is the inverse of $f (x) = \sqrt{4 x + 12}$ .

$f^{- 1} (x) = \frac{x^{2}}{4} - 3$