Trigonometry Examples

$y = \frac{\sqrt{2 x + 4}}{3}$

Step 1

Interchange the variables.

$x = \frac{\sqrt{2 y + 4}}{3}$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\frac{\sqrt{2 y + 4}}{3} = x$ .

$\frac{\sqrt{2 y + 4}}{3} = x$

Step 2.2

Multiply both sides by $3$ .

$\frac{\sqrt{2 y + 4}}{3} \cdot 3 = x \cdot 3$

Step 2.3

Simplify.

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Step 2.3.1

Simplify the left side.

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Step 2.3.1.1

Simplify $\frac{\sqrt{2 y + 4}}{3} \cdot 3$ .

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Step 2.3.1.1.1

Factor $2$ out of $2 y + 4$ .

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Step 2.3.1.1.1.1

Factor $2$ out of $2 y$ .

$\frac{\sqrt{2 (y) + 4}}{3} \cdot 3 = x \cdot 3$

Step 2.3.1.1.1.2

Factor $2$ out of $4$ .

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

Step 2.3.1.1.1.3

Factor $2$ out of $2 y + 2 \cdot 2$ .

$\frac{\sqrt{2 (y + 2)}}{3} \cdot 3 = x \cdot 3$

Step 2.3.1.1.2

Cancel the common factor of $3$ .

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Step 2.3.1.1.2.1

Cancel the common factor.

$\frac{\sqrt{2 (y + 2)}}{3} \cdot 3 = x \cdot 3$

Step 2.3.1.1.2.2

Rewrite the expression.

$\sqrt{2 (y + 2)} = x \cdot 3$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Move $3$ to the left of $x$ .

$\sqrt{2 (y + 2)} = 3 x$

Step 2.4

Solve for $y$ .

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Step 2.4.1

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

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

Step 2.4.2

Simplify each side of the equation.

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Step 2.4.2.1

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

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

Step 2.4.2.2

Simplify the left side.

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Step 2.4.2.2.1

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

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Step 2.4.2.2.1.1

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

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Step 2.4.2.2.1.1.1

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

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

Step 2.4.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.4.2.2.1.1.2.1

Cancel the common factor.

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

Step 2.4.2.2.1.1.2.2

Rewrite the expression.

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

Step 2.4.2.2.1.2

Apply the distributive property.

${(2 y + 2 \cdot 2)}^{1} = {(3 x)}^{2}$

Step 2.4.2.2.1.3

Multiply.

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Step 2.4.2.2.1.3.1

Multiply $2$ by $2$ .

${(2 y + 4)}^{1} = {(3 x)}^{2}$

Step 2.4.2.2.1.3.2

Simplify.

$2 y + 4 = {(3 x)}^{2}$

Step 2.4.2.3

Simplify the right side.

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Step 2.4.2.3.1

Simplify ${(3 x)}^{2}$ .

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Step 2.4.2.3.1.1

Apply the product rule to $3 x$ .

$2 y + 4 = 3^{2} x^{2}$

Step 2.4.2.3.1.2

Raise $3$ to the power of $2$ .

$2 y + 4 = 9 x^{2}$

Step 2.4.3

Solve for $y$ .

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Step 2.4.3.1

Subtract $4$ from both sides of the equation.

$2 y = 9 x^{2} - 4$

Step 2.4.3.2

Divide each term in $2 y = 9 x^{2} - 4$ by $2$ and simplify.

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Step 2.4.3.2.1

Divide each term in $2 y = 9 x^{2} - 4$ by $2$ .

$\frac{2 y}{2} = \frac{9 x^{2}}{2} + \frac{- 4}{2}$

Step 2.4.3.2.2

Simplify the left side.

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Step 2.4.3.2.2.1

Cancel the common factor of $2$ .

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Step 2.4.3.2.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{9 x^{2}}{2} + \frac{- 4}{2}$

Step 2.4.3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{9 x^{2}}{2} + \frac{- 4}{2}$

Step 2.4.3.2.3

Simplify the right side.

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Step 2.4.3.2.3.1

Divide $- 4$ by $2$ .

$y = \frac{9 x^{2}}{2} - 2$

Step 3

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

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

Step 4

Verify if $f^{- 1} (x) = \frac{9 x^{2}}{2} - 2$ is the inverse of $f (x) = \frac{\sqrt{2 x + 4}}{3}$ .

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Step 4.1

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

Step 4.2

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

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Step 4.2.1

Set up the composite result function.

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

Step 4.2.2

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

$f^{- 1} (\frac{\sqrt{2 x + 4}}{3}) = \frac{9 {(\frac{\sqrt{2 x + 4}}{3})}^{2}}{2} - 2$

Step 4.2.3

Simplify each term.

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Step 4.2.3.1

Factor $2$ out of $2 x + 4$ .

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Step 4.2.3.1.1

Factor $2$ out of $2 x$ .

$f^{- 1} (\frac{\sqrt{2 x + 4}}{3}) = \frac{9 {(\frac{\sqrt{2 (x) + 4}}{3})}^{2}}{2} - 2$

Step 4.2.3.1.2

Factor $2$ out of $4$ .

$f^{- 1} (\frac{\sqrt{2 x + 4}}{3}) = \frac{9 {(\frac{\sqrt{2 x + 2 \cdot 2}}{3})}^{2}}{2} - 2$

Step 4.2.3.1.3

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

$f^{- 1} (\frac{\sqrt{2 x + 4}}{3}) = \frac{9 {(\frac{\sqrt{2 (x + 2)}}{3})}^{2}}{2} - 2$

Step 4.2.3.2

Simplify the numerator.

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Step 4.2.3.2.1

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

$f^{- 1} (\frac{\sqrt{2 x + 4}}{3}) = \frac{9 (\frac{{\sqrt{2 (x + 2)}}^{2}}{3^{2}})}{2} - 2$

Step 4.2.3.2.2

Simplify the numerator.

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Step 4.2.3.2.2.1

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

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Step 4.2.3.2.2.1.1

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

$f^{- 1} (\frac{\sqrt{2 x + 4}}{3}) = \frac{9 (\frac{{({(2 (x + 2))}^{\frac{1}{2}})}^{2}}{3^{2}})}{2} - 2$

Step 4.2.3.2.2.1.2

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

$f^{- 1} (\frac{\sqrt{2 x + 4}}{3}) = \frac{9 (\frac{{(2 (x + 2))}^{\frac{1}{2} \cdot 2}}{3^{2}})}{2} - 2$

Step 4.2.3.2.2.1.3

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

$f^{- 1} (\frac{\sqrt{2 x + 4}}{3}) = \frac{9 (\frac{{(2 (x + 2))}^{\frac{2}{2}}}{3^{2}})}{2} - 2$

Step 4.2.3.2.2.1.4

Cancel the common factor of $2$ .

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Step 4.2.3.2.2.1.4.1

Cancel the common factor.

$f^{- 1} (\frac{\sqrt{2 x + 4}}{3}) = \frac{9 (\frac{{(2 (x + 2))}^{\frac{2}{2}}}{3^{2}})}{2} - 2$

Step 4.2.3.2.2.1.4.2

Rewrite the expression.

$f^{- 1} (\frac{\sqrt{2 x + 4}}{3}) = \frac{9 (\frac{2 (x + 2)}{3^{2}})}{2} - 2$

Step 4.2.3.2.2.1.5

Simplify.

$f^{- 1} (\frac{\sqrt{2 x + 4}}{3}) = \frac{9 (\frac{2 (x + 2)}{3^{2}})}{2} - 2$

Step 4.2.3.2.2.2

Apply the distributive property.

$f^{- 1} (\frac{\sqrt{2 x + 4}}{3}) = \frac{9 (\frac{2 x + 2 \cdot 2}{3^{2}})}{2} - 2$

Step 4.2.3.2.2.3

Multiply $2$ by $2$ .

$f^{- 1} (\frac{\sqrt{2 x + 4}}{3}) = \frac{9 (\frac{2 x + 4}{3^{2}})}{2} - 2$

Step 4.2.3.2.2.4

Factor $2$ out of $2 x + 4$ .

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Step 4.2.3.2.2.4.1

Factor $2$ out of $2 x$ .

$f^{- 1} (\frac{\sqrt{2 x + 4}}{3}) = \frac{9 (\frac{2 (x) + 4}{3^{2}})}{2} - 2$

Step 4.2.3.2.2.4.2

Factor $2$ out of $4$ .

$f^{- 1} (\frac{\sqrt{2 x + 4}}{3}) = \frac{9 (\frac{2 x + 2 \cdot 2}{3^{2}})}{2} - 2$

Step 4.2.3.2.2.4.3

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

$f^{- 1} (\frac{\sqrt{2 x + 4}}{3}) = \frac{9 (\frac{2 (x + 2)}{3^{2}})}{2} - 2$

Step 4.2.3.2.3

Raise $3$ to the power of $2$ .

$f^{- 1} (\frac{\sqrt{2 x + 4}}{3}) = \frac{9 (\frac{2 (x + 2)}{9})}{2} - 2$

Step 4.2.3.3

Combine $9$ and $\frac{2 (x + 2)}{9}$ .

$f^{- 1} (\frac{\sqrt{2 x + 4}}{3}) = \frac{\frac{9 (2 (x + 2))}{9}}{2} - 2$

Step 4.2.3.4

Multiply $9$ by $2$ .

$f^{- 1} (\frac{\sqrt{2 x + 4}}{3}) = \frac{\frac{18 (x + 2)}{9}}{2} - 2$

Step 4.2.3.5

Reduce the expression by cancelling the common factors.

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Step 4.2.3.5.1

Reduce the expression $\frac{18 (x + 2)}{9}$ by cancelling the common factors.

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Step 4.2.3.5.1.1

Factor $9$ out of $18 (x + 2)$ .

$f^{- 1} (\frac{\sqrt{2 x + 4}}{3}) = \frac{\frac{9 (2 (x + 2))}{9}}{2} - 2$

Step 4.2.3.5.1.2

Factor $9$ out of $9$ .

$f^{- 1} (\frac{\sqrt{2 x + 4}}{3}) = \frac{\frac{9 (2 (x + 2))}{9 (1)}}{2} - 2$

Step 4.2.3.5.1.3

Cancel the common factor.

$f^{- 1} (\frac{\sqrt{2 x + 4}}{3}) = \frac{\frac{9 (2 (x + 2))}{9 \cdot 1}}{2} - 2$

Step 4.2.3.5.1.4

Rewrite the expression.

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

Step 4.2.3.5.2

Divide $2 (x + 2)$ by $1$ .

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

Step 4.2.3.6

Cancel the common factor of $2$ .

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Step 4.2.3.6.1

Cancel the common factor.

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

Step 4.2.3.6.2

Divide $x + 2$ by $1$ .

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

Step 4.2.4

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

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Step 4.2.4.1

Subtract $2$ from $2$ .

$f^{- 1} (\frac{\sqrt{2 x + 4}}{3}) = x + 0$

Step 4.2.4.2

Add $x$ and $0$ .

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

Step 4.3

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

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Step 4.3.1

Set up the composite result function.

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

Step 4.3.2

Evaluate $f (\frac{9 x^{2}}{2} - 2)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{9 x^{2}}{2} - 2) = \frac{\sqrt{2 (\frac{9 x^{2}}{2} - 2) + 4}}{3}$

Step 4.3.3

Simplify the numerator.

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Step 4.3.3.1

Factor $2$ out of $2 (\frac{9 x^{2}}{2} - 2) + 4$ .

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Step 4.3.3.1.1

Factor $2$ out of $4$ .

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

Step 4.3.3.1.2

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

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

Step 4.3.3.2

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 4.3.3.3

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

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

Step 4.3.3.4

Combine the numerators over the common denominator.

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

Step 4.3.3.5

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 4.3.3.6

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

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

Step 4.3.3.7

Combine the numerators over the common denominator.

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

Step 4.3.3.8

Reorder terms.

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

Step 4.3.3.9

Rewrite $\frac{9 x^{2} + 2 \cdot 2 - 2 \cdot 2}{2}$ in a factored form.

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Step 4.3.3.9.1

Multiply $2$ by $2$ .

$f (\frac{9 x^{2}}{2} - 2) = \frac{\sqrt{2 (\frac{9 x^{2} + 4 - 2 \cdot 2}{2})}}{3}$

Step 4.3.3.9.2

Multiply $- 2$ by $2$ .

$f (\frac{9 x^{2}}{2} - 2) = \frac{\sqrt{2 (\frac{9 x^{2} + 4 - 4}{2})}}{3}$

Step 4.3.3.9.3

Subtract $4$ from $4$ .

$f (\frac{9 x^{2}}{2} - 2) = \frac{\sqrt{2 (\frac{9 x^{2} + 0}{2})}}{3}$

Step 4.3.3.9.4

Add $9 x^{2}$ and $0$ .

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

Step 4.3.3.10

Combine $2$ and $\frac{9 x^{2}}{2}$ .

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

Step 4.3.3.11

Multiply $2$ by $9$ .

$f (\frac{9 x^{2}}{2} - 2) = \frac{\sqrt{\frac{18 x^{2}}{2}}}{3}$

Step 4.3.3.12

Reduce the expression by cancelling the common factors.

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Step 4.3.3.12.1

Reduce the expression $\frac{18 x^{2}}{2}$ by cancelling the common factors.

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Step 4.3.3.12.1.1

Factor $2$ out of $18 x^{2}$ .

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

Step 4.3.3.12.1.2

Factor $2$ out of $2$ .

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

Step 4.3.3.12.1.3

Cancel the common factor.

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

Step 4.3.3.12.1.4

Rewrite the expression.

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

Step 4.3.3.12.2

Divide $9 x^{2}$ by $1$ .

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

Step 4.3.3.13

Rewrite $9 x^{2}$ as ${(3 x)}^{2}$ .

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

Step 4.3.3.14

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

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

Step 4.3.4

Cancel the common factor of $3$ .

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Step 4.3.4.1

Cancel the common factor.

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

Step 4.3.4.2

Divide $x$ by $1$ .

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

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{9 x^{2}}{2} - 2$ is the inverse of $f (x) = \frac{\sqrt{2 x + 4}}{3}$ .

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