Precalculus Examples

$f (x) = \sqrt{3 x - 6}$

Step 1

Write $f (x) = \sqrt{3 x - 6}$ as an equation.

$y = \sqrt{3 x - 6}$

Step 2

Interchange the variables.

$x = \sqrt{3 y - 6}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\sqrt{3 y - 6} = x$ .

$\sqrt{3 y - 6} = x$

Step 3.2

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

${\sqrt{3 y - 6}}^{2} = x^{2}$

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 $\sqrt{3 y - 6}$ as ${(3 y - 6)}^{\frac{1}{2}}$ .

${({(3 y - 6)}^{\frac{1}{2}})}^{2} = x^{2}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Simplify ${({(3 y - 6)}^{\frac{1}{2}})}^{2}$ .

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Step 3.3.2.1.1

Multiply the exponents in ${({(3 y - 6)}^{\frac{1}{2}})}^{2}$ .

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Step 3.3.2.1.1.1

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

${(3 y - 6)}^{\frac{1}{2} \cdot 2} = x^{2}$

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.

${(3 y - 6)}^{\frac{1}{2} \cdot 2} = x^{2}$

Step 3.3.2.1.1.2.2

Rewrite the expression.

${(3 y - 6)}^{1} = x^{2}$

Step 3.3.2.1.2

Simplify.

$3 y - 6 = x^{2}$

Step 3.4

Solve for $y$ .

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Step 3.4.1

Add $6$ to both sides of the equation.

$3 y = x^{2} + 6$

Step 3.4.2

Divide each term in $3 y = x^{2} + 6$ by $3$ and simplify.

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Step 3.4.2.1

Divide each term in $3 y = x^{2} + 6$ by $3$ .

$\frac{3 y}{3} = \frac{x^{2}}{3} + \frac{6}{3}$

Step 3.4.2.2

Simplify the left side.

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Step 3.4.2.2.1

Cancel the common factor of $3$ .

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Step 3.4.2.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{x^{2}}{3} + \frac{6}{3}$

Step 3.4.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{x^{2}}{3} + \frac{6}{3}$

Step 3.4.2.3

Simplify the right side.

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Step 3.4.2.3.1

Divide $6$ by $3$ .

$y = \frac{x^{2}}{3} + 2$

Step 4

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

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

Step 5

Verify if $f^{- 1} (x) = \frac{x^{2}}{3} + 2$ is the inverse of $f (x) = \sqrt{3 x - 6}$ .

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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 $f^{- 1} (\sqrt{3 x - 6})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\sqrt{3 x - 6}) = \frac{{(\sqrt{3 x - 6})}^{2}}{3} + 2$

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 $3$ out of $3 x - 6$ .

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Step 5.2.3.1.1.1

Factor $3$ out of $3 x$ .

$f^{- 1} (\sqrt{3 x - 6}) = \frac{{\sqrt{3 (x) - 6}}^{2}}{3} + 2$

Step 5.2.3.1.1.2

Factor $3$ out of $- 6$ .

$f^{- 1} (\sqrt{3 x - 6}) = \frac{{\sqrt{3 x + 3 \cdot - 2}}^{2}}{3} + 2$

Step 5.2.3.1.1.3

Factor $3$ out of $3 x + 3 \cdot - 2$ .

$f^{- 1} (\sqrt{3 x - 6}) = \frac{{\sqrt{3 (x - 2)}}^{2}}{3} + 2$

Step 5.2.3.1.2

Rewrite ${\sqrt{3 (x - 2)}}^{2}$ as $3 (x - 2)$ .

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Step 5.2.3.1.2.1

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

$f^{- 1} (\sqrt{3 x - 6}) = \frac{{({(3 (x - 2))}^{\frac{1}{2}})}^{2}}{3} + 2$

Step 5.2.3.1.2.2

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

$f^{- 1} (\sqrt{3 x - 6}) = \frac{{(3 (x - 2))}^{\frac{1}{2} \cdot 2}}{3} + 2$

Step 5.2.3.1.2.3

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

$f^{- 1} (\sqrt{3 x - 6}) = \frac{{(3 (x - 2))}^{\frac{2}{2}}}{3} + 2$

Step 5.2.3.1.2.4

Cancel the common factor of $2$ .

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Step 5.2.3.1.2.4.1

Cancel the common factor.

$f^{- 1} (\sqrt{3 x - 6}) = \frac{{(3 (x - 2))}^{\frac{2}{2}}}{3} + 2$

Step 5.2.3.1.2.4.2

Rewrite the expression.

$f^{- 1} (\sqrt{3 x - 6}) = \frac{3 (x - 2)}{3} + 2$

Step 5.2.3.1.2.5

Simplify.

$f^{- 1} (\sqrt{3 x - 6}) = \frac{3 (x - 2)}{3} + 2$

Step 5.2.3.1.3

Apply the distributive property.

$f^{- 1} (\sqrt{3 x - 6}) = \frac{3 x + 3 \cdot - 2}{3} + 2$

Step 5.2.3.1.4

Multiply $3$ by $- 2$ .

$f^{- 1} (\sqrt{3 x - 6}) = \frac{3 x - 6}{3} + 2$

Step 5.2.3.1.5

Factor $3$ out of $3 x - 6$ .

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Step 5.2.3.1.5.1

Factor $3$ out of $3 x$ .

$f^{- 1} (\sqrt{3 x - 6}) = \frac{3 (x) - 6}{3} + 2$

Step 5.2.3.1.5.2

Factor $3$ out of $- 6$ .

$f^{- 1} (\sqrt{3 x - 6}) = \frac{3 x + 3 \cdot - 2}{3} + 2$

Step 5.2.3.1.5.3

Factor $3$ out of $3 x + 3 \cdot - 2$ .

$f^{- 1} (\sqrt{3 x - 6}) = \frac{3 (x - 2)}{3} + 2$

Step 5.2.3.2

Cancel the common factor of $3$ .

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Step 5.2.3.2.1

Cancel the common factor.

$f^{- 1} (\sqrt{3 x - 6}) = \frac{3 (x - 2)}{3} + 2$

Step 5.2.3.2.2

Divide $x - 2$ by $1$ .

$f^{- 1} (\sqrt{3 x - 6}) = x - 2 + 2$

Step 5.2.4

Combine the opposite terms in $x - 2 + 2$ .

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Step 5.2.4.1

Add $- 2$ and $2$ .

$f^{- 1} (\sqrt{3 x - 6}) = x + 0$

Step 5.2.4.2

Add $x$ and $0$ .

$f^{- 1} (\sqrt{3 x - 6}) = x$

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

$f (\frac{x^{2}}{3} + 2) = \sqrt{3 (\frac{x^{2}}{3} + 2) - 6}$

Step 5.3.3

Factor $3$ out of $3 (\frac{x^{2}}{3} + 2) - 6$ .

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Step 5.3.3.1

Factor $3$ out of $- 6$ .

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

Step 5.3.3.2

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

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

Step 5.3.4

Simplify by subtracting numbers.

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Step 5.3.4.1

Subtract $2$ from $2$ .

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

Step 5.3.4.2

Add $\frac{x^{2}}{3}$ and $0$ .

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

Step 5.3.5

Combine $3$ and $\frac{x^{2}}{3}$ .

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

Step 5.3.6

Reduce the expression by cancelling the common factors.

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Step 5.3.6.1

Reduce the expression $\frac{3 x^{2}}{3}$ by cancelling the common factors.

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Step 5.3.6.1.1

Cancel the common factor.

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

Step 5.3.6.1.2

Rewrite the expression.

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

Step 5.3.6.2

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

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

Step 5.3.7

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

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

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x^{2}}{3} + 2$ is the inverse of $f (x) = \sqrt{3 x - 6}$ .

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