Trigonometry Examples

$f (x) = \frac{8}{27} \cdot (x (3))$

Step 1

Write $f (x) = \frac{8}{27} \cdot (x (3))$ as an equation.

$y = \frac{8}{27} \cdot (x (3))$

Step 2

Interchange the variables.

$x = \frac{8}{27} \cdot (y (3))$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{8}{27} y \cdot 3 = x$ .

$\frac{8}{27} y \cdot 3 = x$

Step 3.2

Simplify $\frac{8}{27} y \cdot 3$ .

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Step 3.2.1

Combine $\frac{8}{27}$ and $y$ .

$\frac{8 y}{27} \cdot 3 = x$

Step 3.2.2

Cancel the common factor of $3$ .

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Step 3.2.2.1

Factor $3$ out of $27$ .

$\frac{8 y}{3 (9)} \cdot 3 = x$

Step 3.2.2.2

Cancel the common factor.

$\frac{8 y}{3 \cdot 9} \cdot 3 = x$

Step 3.2.2.3

Rewrite the expression.

$\frac{8 y}{9} = x$

Step 3.3

Multiply both sides of the equation by $\frac{9}{8}$ .

$\frac{9}{8} \cdot \frac{8 y}{9} = \frac{9}{8} x$

Step 3.4

Simplify both sides of the equation.

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Step 3.4.1

Simplify the left side.

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Step 3.4.1.1

Simplify $\frac{9}{8} \cdot \frac{8 y}{9}$ .

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Step 3.4.1.1.1

Cancel the common factor of $9$ .

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Step 3.4.1.1.1.1

Cancel the common factor.

$\frac{9}{8} \cdot \frac{8 y}{9} = \frac{9}{8} x$

Step 3.4.1.1.1.2

Rewrite the expression.

$\frac{1}{8} (8 y) = \frac{9}{8} x$

Step 3.4.1.1.2

Cancel the common factor of $8$ .

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Step 3.4.1.1.2.1

Factor $8$ out of $8 y$ .

$\frac{1}{8} (8 (y)) = \frac{9}{8} x$

Step 3.4.1.1.2.2

Cancel the common factor.

$\frac{1}{8} (8 y) = \frac{9}{8} x$

Step 3.4.1.1.2.3

Rewrite the expression.

$y = \frac{9}{8} x$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Combine $\frac{9}{8}$ and $x$ .

$y = \frac{9 x}{8}$

Step 4

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

$f^{- 1} (x) = \frac{9 x}{8}$

Step 5

Verify if $f^{- 1} (x) = \frac{9 x}{8}$ is the inverse of $f (x) = \frac{8}{27} \cdot (x (3))$ .

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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 $f^{- 1} (\frac{8}{27} \cdot (x (3)))$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{8}{27} \cdot (x (3))) = \frac{9 (\frac{8}{27} \cdot (x (3)))}{8}$

Step 5.2.3

Simplify the numerator.

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Step 5.2.3.1

Combine $9$ and $\frac{8}{27}$ .

$f^{- 1} (\frac{8}{27} \cdot (x (3))) = \frac{\frac{9 \cdot 8}{27} x \cdot 3}{8}$

Step 5.2.3.2

Combine $\frac{9 \cdot 8}{27}$ and $x$ .

$f^{- 1} (\frac{8}{27} \cdot (x (3))) = \frac{\frac{9 \cdot (8 x)}{27} \cdot 3}{8}$

Step 5.2.3.3

Combine $\frac{9 \cdot 8 x}{27}$ and $3$ .

$f^{- 1} (\frac{8}{27} \cdot (x (3))) = \frac{\frac{9 \cdot (8 x) \cdot 3}{27}}{8}$

Step 5.2.4

Simplify the numerator.

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Step 5.2.4.1

Multiply $9$ by $8$ .

$f^{- 1} (\frac{8}{27} \cdot (x (3))) = \frac{\frac{72 x \cdot 3}{27}}{8}$

Step 5.2.4.2

Multiply $3$ by $72$ .

$f^{- 1} (\frac{8}{27} \cdot (x (3))) = \frac{\frac{216 x}{27}}{8}$

Step 5.2.5

Reduce the expression by cancelling the common factors.

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Step 5.2.5.1

Reduce the expression $\frac{216 x}{27}$ by cancelling the common factors.

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Step 5.2.5.1.1

Factor $27$ out of $216 x$ .

$f^{- 1} (\frac{8}{27} \cdot (x (3))) = \frac{\frac{27 (8 x)}{27}}{8}$

Step 5.2.5.1.2

Factor $27$ out of $27$ .

$f^{- 1} (\frac{8}{27} \cdot (x (3))) = \frac{\frac{27 (8 x)}{27 (1)}}{8}$

Step 5.2.5.1.3

Cancel the common factor.

$f^{- 1} (\frac{8}{27} \cdot (x (3))) = \frac{\frac{27 (8 x)}{27 \cdot 1}}{8}$

Step 5.2.5.1.4

Rewrite the expression.

$f^{- 1} (\frac{8}{27} \cdot (x (3))) = \frac{\frac{8 x}{1}}{8}$

Step 5.2.5.2

Divide $8 x$ by $1$ .

$f^{- 1} (\frac{8}{27} \cdot (x (3))) = \frac{8 x}{8}$

Step 5.2.6

Cancel the common factor of $8$ .

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Step 5.2.6.1

Cancel the common factor.

$f^{- 1} (\frac{8}{27} \cdot (x (3))) = \frac{8 x}{8}$

Step 5.2.6.2

Divide $x$ by $1$ .

$f^{- 1} (\frac{8}{27} \cdot (x (3))) = x$

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 $f (\frac{9 x}{8})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{9 x}{8}) = \frac{8}{27} \cdot ((\frac{9 x}{8}) (3))$

Step 5.3.3

Cancel the common factor of $3$ .

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Step 5.3.3.1

Factor $3$ out of $27$ .

$f (\frac{9 x}{8}) = \frac{8}{3 (9)} \cdot ((\frac{9 x}{8}) (3))$

Step 5.3.3.2

Factor $3$ out of $(\frac{9 x}{8}) (3)$ .

$f (\frac{9 x}{8}) = \frac{8}{3 (9)} \cdot (3 ((\frac{3 x}{8}) \cdot 3))$

Step 5.3.3.3

Cancel the common factor.

$f (\frac{9 x}{8}) = \frac{8}{3 \cdot 9} \cdot (3 ((\frac{3 x}{8}) \cdot 3))$

Step 5.3.3.4

Rewrite the expression.

$f (\frac{9 x}{8}) = \frac{8}{9} \cdot ((\frac{3 x}{8}) \cdot 3)$

Step 5.3.4

Combine $\frac{3 x}{8}$ and $3$ .

$f (\frac{9 x}{8}) = \frac{8}{9} \cdot \frac{3 x \cdot 3}{8}$

Step 5.3.5

Multiply $3$ by $3$ .

$f (\frac{9 x}{8}) = \frac{8}{9} \cdot \frac{9 x}{8}$

Step 5.3.6

Multiply $\frac{8}{9}$ by $\frac{9 x}{8}$ .

$f (\frac{9 x}{8}) = \frac{8 (9 x)}{9 \cdot 8}$

Step 5.3.7

Multiply.

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Step 5.3.7.1

Multiply $9$ by $8$ .

$f (\frac{9 x}{8}) = \frac{72 x}{9 \cdot 8}$

Step 5.3.7.2

Multiply $9$ by $8$ .

$f (\frac{9 x}{8}) = \frac{72 x}{72}$

Step 5.3.8

Cancel the common factor of $72$ .

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Step 5.3.8.1

Cancel the common factor.

$f (\frac{9 x}{8}) = \frac{72 x}{72}$

Step 5.3.8.2

Divide $x$ by $1$ .

$f (\frac{9 x}{8}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{9 x}{8}$ is the inverse of $f (x) = \frac{8}{27} \cdot (x (3))$ .

$f^{- 1} (x) = \frac{9 x}{8}$