Trigonometry Examples

$f (x) = \sqrt{6 x - 9}$

Step 1

Write $f (x) = \sqrt{6 x - 9}$ as an equation.

$y = \sqrt{6 x - 9}$

Step 2

Interchange the variables.

$x = \sqrt{6 y - 9}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\sqrt{6 y - 9} = x$ .

$\sqrt{6 y - 9} = x$

Step 3.2

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

${\sqrt{6 y - 9}}^{2} = x^{2}$

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 $\sqrt{6 y - 9}$ as ${(6 y - 9)}^{\frac{1}{2}}$ .

${({(6 y - 9)}^{\frac{1}{2}})}^{2} = x^{2}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Simplify ${({(6 y - 9)}^{\frac{1}{2}})}^{2}$ .

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Step 3.3.2.1.1

Multiply the exponents in ${({(6 y - 9)}^{\frac{1}{2}})}^{2}$ .

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Step 3.3.2.1.1.1

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

${(6 y - 9)}^{\frac{1}{2} \cdot 2} = x^{2}$

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.

${(6 y - 9)}^{\frac{1}{2} \cdot 2} = x^{2}$

Step 3.3.2.1.1.2.2

Rewrite the expression.

${(6 y - 9)}^{1} = x^{2}$

Step 3.3.2.1.2

Simplify.

$6 y - 9 = x^{2}$

Step 3.4

Solve for $y$ .

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Step 3.4.1

Add $9$ to both sides of the equation.

$6 y = x^{2} + 9$

Step 3.4.2

Divide each term in $6 y = x^{2} + 9$ by $6$ and simplify.

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Step 3.4.2.1

Divide each term in $6 y = x^{2} + 9$ by $6$ .

$\frac{6 y}{6} = \frac{x^{2}}{6} + \frac{9}{6}$

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.

$\frac{6 y}{6} = \frac{x^{2}}{6} + \frac{9}{6}$

Step 3.4.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{x^{2}}{6} + \frac{9}{6}$

Step 3.4.2.3

Simplify the right side.

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Step 3.4.2.3.1

Cancel the common factor of $9$ and $6$ .

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Step 3.4.2.3.1.1

Factor $3$ out of $9$ .

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

Step 3.4.2.3.1.2

Cancel the common factors.

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Step 3.4.2.3.1.2.1

Factor $3$ out of $6$ .

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

Step 3.4.2.3.1.2.2

Cancel the common factor.

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

Step 3.4.2.3.1.2.3

Rewrite the expression.

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

Step 4

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

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

Step 5

Verify if $f^{- 1} (x) = \frac{x^{2}}{6} + \frac{3}{2}$ is the inverse of $f (x) = \sqrt{6 x - 9}$ .

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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 $f^{- 1} (\sqrt{6 x - 9})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\sqrt{6 x - 9}) = \frac{{(\sqrt{6 x - 9})}^{2}}{6} + \frac{3}{2}$

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 $6 x - 9$ .

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Step 5.2.3.1.1.1

Factor $3$ out of $6 x$ .

$f^{- 1} (\sqrt{6 x - 9}) = \frac{{\sqrt{3 (2 x) - 9}}^{2}}{6} + \frac{3}{2}$

Step 5.2.3.1.1.2

Factor $3$ out of $- 9$ .

$f^{- 1} (\sqrt{6 x - 9}) = \frac{{\sqrt{3 (2 x) + 3 (- 3)}}^{2}}{6} + \frac{3}{2}$

Step 5.2.3.1.1.3

Factor $3$ out of $3 (2 x) + 3 (- 3)$ .

$f^{- 1} (\sqrt{6 x - 9}) = \frac{{\sqrt{3 (2 x - 3)}}^{2}}{6} + \frac{3}{2}$

Step 5.2.3.1.2

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

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Step 5.2.3.1.2.1

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

$f^{- 1} (\sqrt{6 x - 9}) = \frac{{({(3 (2 x - 3))}^{\frac{1}{2}})}^{2}}{6} + \frac{3}{2}$

Step 5.2.3.1.2.2

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

$f^{- 1} (\sqrt{6 x - 9}) = \frac{{(3 (2 x - 3))}^{\frac{1}{2} \cdot 2}}{6} + \frac{3}{2}$

Step 5.2.3.1.2.3

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

$f^{- 1} (\sqrt{6 x - 9}) = \frac{{(3 (2 x - 3))}^{\frac{2}{2}}}{6} + \frac{3}{2}$

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.

$f^{- 1} (\sqrt{6 x - 9}) = \frac{{(3 (2 x - 3))}^{\frac{2}{2}}}{6} + \frac{3}{2}$

Step 5.2.3.1.2.4.2

Rewrite the expression.

$f^{- 1} (\sqrt{6 x - 9}) = \frac{3 (2 x - 3)}{6} + \frac{3}{2}$

Step 5.2.3.1.2.5

Simplify.

$f^{- 1} (\sqrt{6 x - 9}) = \frac{3 (2 x - 3)}{6} + \frac{3}{2}$

Step 5.2.3.1.3

Apply the distributive property.

$f^{- 1} (\sqrt{6 x - 9}) = \frac{3 (2 x) + 3 \cdot - 3}{6} + \frac{3}{2}$

Step 5.2.3.1.4

Multiply $2$ by $3$ .

$f^{- 1} (\sqrt{6 x - 9}) = \frac{6 x + 3 \cdot - 3}{6} + \frac{3}{2}$

Step 5.2.3.1.5

Multiply $3$ by $- 3$ .

$f^{- 1} (\sqrt{6 x - 9}) = \frac{6 x - 9}{6} + \frac{3}{2}$

Step 5.2.3.1.6

Factor $3$ out of $6 x - 9$ .

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Step 5.2.3.1.6.1

Factor $3$ out of $6 x$ .

$f^{- 1} (\sqrt{6 x - 9}) = \frac{3 (2 x) - 9}{6} + \frac{3}{2}$

Step 5.2.3.1.6.2

Factor $3$ out of $- 9$ .

$f^{- 1} (\sqrt{6 x - 9}) = \frac{3 (2 x) + 3 (- 3)}{6} + \frac{3}{2}$

Step 5.2.3.1.6.3

Factor $3$ out of $3 (2 x) + 3 (- 3)$ .

$f^{- 1} (\sqrt{6 x - 9}) = \frac{3 (2 x - 3)}{6} + \frac{3}{2}$

Step 5.2.3.2

Cancel the common factors.

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Step 5.2.3.2.1

Factor $3$ out of $6$ .

$f^{- 1} (\sqrt{6 x - 9}) = \frac{3 (2 x - 3)}{3 \cdot 2} + \frac{3}{2}$

Step 5.2.3.2.2

Cancel the common factor.

$f^{- 1} (\sqrt{6 x - 9}) = \frac{3 (2 x - 3)}{3 \cdot 2} + \frac{3}{2}$

Step 5.2.3.2.3

Rewrite the expression.

$f^{- 1} (\sqrt{6 x - 9}) = \frac{2 x - 3}{2} + \frac{3}{2}$

Step 5.2.4

Simplify terms.

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Step 5.2.4.1

Combine the numerators over the common denominator.

$f^{- 1} (\sqrt{6 x - 9}) = \frac{2 x - 3 + 3}{2}$

Step 5.2.4.2

Combine the opposite terms in $2 x - 3 + 3$ .

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Step 5.2.4.2.1

Add $- 3$ and $3$ .

$f^{- 1} (\sqrt{6 x - 9}) = \frac{2 x + 0}{2}$

Step 5.2.4.2.2

Add $2 x$ and $0$ .

$f^{- 1} (\sqrt{6 x - 9}) = \frac{2 x}{2}$

Step 5.2.4.3

Cancel the common factor of $2$ .

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Step 5.2.4.3.1

Cancel the common factor.

$f^{- 1} (\sqrt{6 x - 9}) = \frac{2 x}{2}$

Step 5.2.4.3.2

Divide $x$ by $1$ .

$f^{- 1} (\sqrt{6 x - 9}) = x$

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 $f (\frac{x^{2}}{6} + \frac{3}{2})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x^{2}}{6} + \frac{3}{2}) = \sqrt{6 (\frac{x^{2}}{6} + \frac{3}{2}) - 9}$

Step 5.3.3

Factor $3$ out of $6 (\frac{x^{2}}{6} + \frac{3}{2}) - 9$ .

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Step 5.3.3.1

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

$f (\frac{x^{2}}{6} + \frac{3}{2}) = \sqrt{3 (2 (\frac{x^{2}}{6} + \frac{3}{2})) - 9}$

Step 5.3.3.2

Factor $3$ out of $- 9$ .

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

Step 5.3.3.3

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

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

Step 5.3.4

Apply the distributive property.

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

Step 5.3.5

Cancel the common factor of $2$ .

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Step 5.3.5.1

Factor $2$ out of $6$ .

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

Step 5.3.5.2

Cancel the common factor.

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

Step 5.3.5.3

Rewrite the expression.

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

Step 5.3.6

Cancel the common factor of $2$ .

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Step 5.3.6.1

Cancel the common factor.

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

Step 5.3.6.2

Rewrite the expression.

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

Step 5.3.7

Simplify by subtracting numbers.

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Step 5.3.7.1

Subtract $3$ from $3$ .

$f (\frac{x^{2}}{6} + \frac{3}{2}) = \sqrt{3 (\frac{x^{2}}{3} + 0)}$

Step 5.3.7.2

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

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

Step 5.3.8

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

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

Step 5.3.9

Reduce the expression by cancelling the common factors.

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Step 5.3.9.1

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

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Step 5.3.9.1.1

Cancel the common factor.

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

Step 5.3.9.1.2

Rewrite the expression.

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

Step 5.3.9.2

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

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

Step 5.3.10

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

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

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x^{2}}{6} + \frac{3}{2}$ is the inverse of $f (x) = \sqrt{6 x - 9}$ .

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