Trigonometry Examples

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

Step 1

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

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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Step 3.1

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

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

Step 3.2

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

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

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

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

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

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

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Step 3.3.2.1.1

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

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Step 3.3.2.1.1.1

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

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

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.

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

Step 3.3.2.1.1.2.2

Rewrite the expression.

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

Step 3.3.2.1.2

Simplify.

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

Step 3.4

Solve for $y$ .

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Step 3.4.1

Add $6$ to both sides of the equation.

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

Step 3.4.2

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

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Step 3.4.2.1

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

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

Step 3.4.2.2

Simplify the left side.

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Step 3.4.2.2.1

Cancel the common factor of $6$ .

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Step 3.4.2.2.1.1

Cancel the common factor.

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

Step 3.4.2.2.1.2

Divide $y$ by $1$ .

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

Step 3.4.2.3

Simplify the right side.

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Step 3.4.2.3.1

Divide $6$ by $6$ .

$y = \frac{x^{2}}{6} + 1$

Step 4

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

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

Step 5

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

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

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

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

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Step 5.2.3.1.1.1

Factor $6$ out of $6 x$ .

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

Step 5.2.3.1.1.2

Factor $6$ out of $- 6$ .

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

Step 5.2.3.1.1.3

Factor $6$ out of $6 (x) + 6 (- 1)$ .

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

Step 5.2.3.1.2

Rewrite ${\sqrt{6 (x - 1)}}^{2}$ as $6 (x - 1)$ .

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Step 5.2.3.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6 (x - 1)}$ as ${(6 (x - 1))}^{\frac{1}{2}}$ .

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

Step 5.2.3.1.2.2

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

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

Step 5.2.3.1.2.3

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

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

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.

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

Step 5.2.3.1.2.4.2

Rewrite the expression.

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

Step 5.2.3.1.2.5

Simplify.

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

Step 5.2.3.1.3

Apply the distributive property.

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

Step 5.2.3.1.4

Multiply $6$ by $- 1$ .

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

Step 5.2.3.1.5

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

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Step 5.2.3.1.5.1

Factor $6$ out of $6 x$ .

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

Step 5.2.3.1.5.2

Factor $6$ out of $- 6$ .

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

Step 5.2.3.1.5.3

Factor $6$ out of $6 (x) + 6 (- 1)$ .

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

Step 5.2.3.2

Cancel the common factor of $6$ .

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Step 5.2.3.2.1

Cancel the common factor.

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

Step 5.2.3.2.2

Divide $x - 1$ by $1$ .

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

Step 5.2.4

Combine the opposite terms in $x - 1 + 1$ .

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Step 5.2.4.1

Add $- 1$ and $1$ .

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

Step 5.2.4.2

Add $x$ and $0$ .

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

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 $f (\frac{x^{2}}{6} + 1)$ by substituting in the value of $f^{- 1}$ into $f$ .

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

Step 5.3.3

Factor $6$ out of $6 (\frac{x^{2}}{6} + 1) - 6$ .

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Step 5.3.3.1

Factor $6$ out of $- 6$ .

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

Step 5.3.3.2

Factor $6$ out of $6 (\frac{x^{2}}{6} + 1) + 6 (- 1)$ .

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

Step 5.3.4

Simplify by subtracting numbers.

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Step 5.3.4.1

Subtract $1$ from $1$ .

$f (\frac{x^{2}}{6} + 1) = \sqrt{6 (\frac{x^{2}}{6} + 0)}$

Step 5.3.4.2

Add $\frac{x^{2}}{6}$ and $0$ .

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

Step 5.3.5

Combine $6$ and $\frac{x^{2}}{6}$ .

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

Step 5.3.6

Reduce the expression by cancelling the common factors.

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Step 5.3.6.1

Reduce the expression $\frac{6 x^{2}}{6}$ by cancelling the common factors.

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Step 5.3.6.1.1

Cancel the common factor.

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

Step 5.3.6.1.2

Rewrite the expression.

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

Step 5.3.6.2

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

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

Step 5.3.7

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

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

Step 5.4

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

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