Trigonometry Examples

$f (x) = \frac{3 x}{5 x - 2}$

Step 1

Write $f (x) = \frac{3 x}{5 x - 2}$ as an equation.

$y = \frac{3 x}{5 x - 2}$

Step 2

Interchange the variables.

$x = \frac{3 y}{5 y - 2}$

Step 3

Solve for $y$ .

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Step 3.1

Multiply the equation by $5 y - 2$ .

$(5 y - 2) (x) = (\frac{3 y}{5 y - 2}) (5 y - 2)$

Step 3.2

Simplify the left side.

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Step 3.2.1

Apply the distributive property.

$5 y x - 2 x = (\frac{3 y}{5 y - 2}) (5 y - 2)$

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of $5 y - 2$ .

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Step 3.3.1.1

Cancel the common factor.

$5 y x - 2 x = \frac{3 y}{5 y - 2} (5 y - 2)$

Step 3.3.1.2

Rewrite the expression.

$5 y x - 2 x = 3 y$

Step 3.4

Solve for $y$ .

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Step 3.4.1

Subtract $3 y$ from both sides of the equation.

$5 y x - 2 x - 3 y = 0$

Step 3.4.2

Add $2 x$ to both sides of the equation.

$5 y x - 3 y = 2 x$

Step 3.4.3

Factor $y$ out of $5 y x - 3 y$ .

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Step 3.4.3.1

Factor $y$ out of $5 y x$ .

$y (5 x) - 3 y = 2 x$

Step 3.4.3.2

Factor $y$ out of $- 3 y$ .

$y (5 x) + y \cdot - 3 = 2 x$

Step 3.4.3.3

Factor $y$ out of $y (5 x) + y \cdot - 3$ .

$y (5 x - 3) = 2 x$

Step 3.4.4

Divide each term in $y (5 x - 3) = 2 x$ by $5 x - 3$ and simplify.

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Step 3.4.4.1

Divide each term in $y (5 x - 3) = 2 x$ by $5 x - 3$ .

$\frac{y (5 x - 3)}{5 x - 3} = \frac{2 x}{5 x - 3}$

Step 3.4.4.2

Simplify the left side.

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Step 3.4.4.2.1

Cancel the common factor of $5 x - 3$ .

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Step 3.4.4.2.1.1

Cancel the common factor.

$\frac{y (5 x - 3)}{5 x - 3} = \frac{2 x}{5 x - 3}$

Step 3.4.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{2 x}{5 x - 3}$

Step 4

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

$f^{- 1} (x) = \frac{2 x}{5 x - 3}$

Step 5

Verify if $f^{- 1} (x) = \frac{2 x}{5 x - 3}$ is the inverse of $f (x) = \frac{3 x}{5 x - 2}$ .

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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 $f^{- 1} (\frac{3 x}{5 x - 2})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{3 x}{5 x - 2}) = \frac{2 (\frac{3 x}{5 x - 2})}{5 (\frac{3 x}{5 x - 2}) - 3}$

Step 5.2.3

Combine $2$ and $\frac{3 x}{5 x - 2}$ .

$f^{- 1} (\frac{3 x}{5 x - 2}) = \frac{\frac{2 (3 x)}{5 x - 2}}{5 (\frac{3 x}{5 x - 2}) - 3}$

Step 5.2.4

Simplify the denominator.

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Step 5.2.4.1

Multiply $5 \frac{3 x}{5 x - 2}$ .

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Step 5.2.4.1.1

Combine $5$ and $\frac{3 x}{5 x - 2}$ .

$f^{- 1} (\frac{3 x}{5 x - 2}) = \frac{\frac{2 (3 x)}{5 x - 2}}{\frac{5 (3 x)}{5 x - 2} - 3}$

Step 5.2.4.1.2

Multiply $3$ by $5$ .

$f^{- 1} (\frac{3 x}{5 x - 2}) = \frac{\frac{2 (3 x)}{5 x - 2}}{\frac{15 x}{5 x - 2} - 3}$

Step 5.2.4.2

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{5 x - 2}{5 x - 2}$ .

$f^{- 1} (\frac{3 x}{5 x - 2}) = \frac{\frac{2 (3 x)}{5 x - 2}}{\frac{15 x}{5 x - 2} - 3 \cdot \frac{5 x - 2}{5 x - 2}}$

Step 5.2.4.3

Combine $- 3$ and $\frac{5 x - 2}{5 x - 2}$ .

$f^{- 1} (\frac{3 x}{5 x - 2}) = \frac{\frac{2 (3 x)}{5 x - 2}}{\frac{15 x}{5 x - 2} + \frac{- 3 (5 x - 2)}{5 x - 2}}$

Step 5.2.4.4

Combine the numerators over the common denominator.

$f^{- 1} (\frac{3 x}{5 x - 2}) = \frac{\frac{2 (3 x)}{5 x - 2}}{\frac{15 x - 3 (5 x - 2)}{5 x - 2}}$

Step 5.2.4.5

Rewrite $\frac{15 x - 3 (5 x - 2)}{5 x - 2}$ in a factored form.

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Step 5.2.4.5.1

Factor $3$ out of $15 x - 3 (5 x - 2)$ .

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Step 5.2.4.5.1.1

Factor $3$ out of $15 x$ .

$f^{- 1} (\frac{3 x}{5 x - 2}) = \frac{\frac{2 (3 x)}{5 x - 2}}{\frac{3 (5 x) - 3 (5 x - 2)}{5 x - 2}}$

Step 5.2.4.5.1.2

Factor $3$ out of $- 3 (5 x - 2)$ .

$f^{- 1} (\frac{3 x}{5 x - 2}) = \frac{\frac{2 (3 x)}{5 x - 2}}{\frac{3 (5 x) + 3 (- (5 x - 2))}{5 x - 2}}$

Step 5.2.4.5.1.3

Factor $3$ out of $3 (5 x) + 3 (- (5 x - 2))$ .

$f^{- 1} (\frac{3 x}{5 x - 2}) = \frac{\frac{2 (3 x)}{5 x - 2}}{\frac{3 (5 x - (5 x - 2))}{5 x - 2}}$

Step 5.2.4.5.2

Apply the distributive property.

$f^{- 1} (\frac{3 x}{5 x - 2}) = \frac{\frac{2 (3 x)}{5 x - 2}}{\frac{3 (5 x - (5 x) + 2)}{5 x - 2}}$

Step 5.2.4.5.3

Multiply $5$ by $- 1$ .

$f^{- 1} (\frac{3 x}{5 x - 2}) = \frac{\frac{2 (3 x)}{5 x - 2}}{\frac{3 (5 x - 5 x + 2)}{5 x - 2}}$

Step 5.2.4.5.4

Multiply $- 1$ by $- 2$ .

$f^{- 1} (\frac{3 x}{5 x - 2}) = \frac{\frac{2 (3 x)}{5 x - 2}}{\frac{3 (5 x - 5 x + 2)}{5 x - 2}}$

Step 5.2.4.5.5

Subtract $5 x$ from $5 x$ .

$f^{- 1} (\frac{3 x}{5 x - 2}) = \frac{\frac{2 (3 x)}{5 x - 2}}{\frac{3 (0 + 2)}{5 x - 2}}$

Step 5.2.4.5.6

Add $0$ and $2$ .

$f^{- 1} (\frac{3 x}{5 x - 2}) = \frac{\frac{2 (3 x)}{5 x - 2}}{\frac{3 \cdot 2}{5 x - 2}}$

Step 5.2.4.6

Multiply $3$ by $2$ .

$f^{- 1} (\frac{3 x}{5 x - 2}) = \frac{\frac{2 (3 x)}{5 x - 2}}{\frac{6}{5 x - 2}}$

Step 5.2.5

Multiply $2$ by $3$ .

$f^{- 1} (\frac{3 x}{5 x - 2}) = \frac{\frac{6 x}{5 x - 2}}{\frac{6}{5 x - 2}}$

Step 5.2.6

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{3 x}{5 x - 2}) = \frac{6 x}{5 x - 2} \cdot \frac{5 x - 2}{6}$

Step 5.2.7

Cancel the common factor of $6$ .

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Step 5.2.7.1

Factor $6$ out of $6 x$ .

$f^{- 1} (\frac{3 x}{5 x - 2}) = \frac{6 (x)}{5 x - 2} \cdot \frac{5 x - 2}{6}$

Step 5.2.7.2

Cancel the common factor.

$f^{- 1} (\frac{3 x}{5 x - 2}) = \frac{6 x}{5 x - 2} \cdot \frac{5 x - 2}{6}$

Step 5.2.7.3

Rewrite the expression.

$f^{- 1} (\frac{3 x}{5 x - 2}) = \frac{x}{5 x - 2} \cdot (5 x - 2)$

Step 5.2.8

Cancel the common factor of $5 x - 2$ .

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Step 5.2.8.1

Cancel the common factor.

$f^{- 1} (\frac{3 x}{5 x - 2}) = \frac{x}{5 x - 2} \cdot (5 x - 2)$

Step 5.2.8.2

Rewrite the expression.

$f^{- 1} (\frac{3 x}{5 x - 2}) = x$

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 $f (\frac{2 x}{5 x - 3})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{2 x}{5 x - 3}) = \frac{3 (\frac{2 x}{5 x - 3})}{5 (\frac{2 x}{5 x - 3}) - 2}$

Step 5.3.3

Combine $3$ and $\frac{2 x}{5 x - 3}$ .

$f (\frac{2 x}{5 x - 3}) = \frac{\frac{3 (2 x)}{5 x - 3}}{5 (\frac{2 x}{5 x - 3}) - 2}$

Step 5.3.4

Simplify the denominator.

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Step 5.3.4.1

Multiply $5 \frac{2 x}{5 x - 3}$ .

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Step 5.3.4.1.1

Combine $5$ and $\frac{2 x}{5 x - 3}$ .

$f (\frac{2 x}{5 x - 3}) = \frac{\frac{3 (2 x)}{5 x - 3}}{\frac{5 (2 x)}{5 x - 3} - 2}$

Step 5.3.4.1.2

Multiply $2$ by $5$ .

$f (\frac{2 x}{5 x - 3}) = \frac{\frac{3 (2 x)}{5 x - 3}}{\frac{10 x}{5 x - 3} - 2}$

Step 5.3.4.2

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{5 x - 3}{5 x - 3}$ .

$f (\frac{2 x}{5 x - 3}) = \frac{\frac{3 (2 x)}{5 x - 3}}{\frac{10 x}{5 x - 3} - 2 \cdot \frac{5 x - 3}{5 x - 3}}$

Step 5.3.4.3

Combine $- 2$ and $\frac{5 x - 3}{5 x - 3}$ .

$f (\frac{2 x}{5 x - 3}) = \frac{\frac{3 (2 x)}{5 x - 3}}{\frac{10 x}{5 x - 3} + \frac{- 2 (5 x - 3)}{5 x - 3}}$

Step 5.3.4.4

Combine the numerators over the common denominator.

$f (\frac{2 x}{5 x - 3}) = \frac{\frac{3 (2 x)}{5 x - 3}}{\frac{10 x - 2 (5 x - 3)}{5 x - 3}}$

Step 5.3.4.5

Rewrite $\frac{10 x - 2 (5 x - 3)}{5 x - 3}$ in a factored form.

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Step 5.3.4.5.1

Factor $2$ out of $10 x - 2 (5 x - 3)$ .

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Step 5.3.4.5.1.1

Factor $2$ out of $10 x$ .

$f (\frac{2 x}{5 x - 3}) = \frac{\frac{3 (2 x)}{5 x - 3}}{\frac{2 (5 x) - 2 (5 x - 3)}{5 x - 3}}$

Step 5.3.4.5.1.2

Factor $2$ out of $- 2 (5 x - 3)$ .

$f (\frac{2 x}{5 x - 3}) = \frac{\frac{3 (2 x)}{5 x - 3}}{\frac{2 (5 x) + 2 (- (5 x - 3))}{5 x - 3}}$

Step 5.3.4.5.1.3

Factor $2$ out of $2 (5 x) + 2 (- (5 x - 3))$ .

$f (\frac{2 x}{5 x - 3}) = \frac{\frac{3 (2 x)}{5 x - 3}}{\frac{2 (5 x - (5 x - 3))}{5 x - 3}}$

Step 5.3.4.5.2

Apply the distributive property.

$f (\frac{2 x}{5 x - 3}) = \frac{\frac{3 (2 x)}{5 x - 3}}{\frac{2 (5 x - (5 x) + 3)}{5 x - 3}}$

Step 5.3.4.5.3

Multiply $5$ by $- 1$ .

$f (\frac{2 x}{5 x - 3}) = \frac{\frac{3 (2 x)}{5 x - 3}}{\frac{2 (5 x - 5 x + 3)}{5 x - 3}}$

Step 5.3.4.5.4

Multiply $- 1$ by $- 3$ .

$f (\frac{2 x}{5 x - 3}) = \frac{\frac{3 (2 x)}{5 x - 3}}{\frac{2 (5 x - 5 x + 3)}{5 x - 3}}$

Step 5.3.4.5.5

Subtract $5 x$ from $5 x$ .

$f (\frac{2 x}{5 x - 3}) = \frac{\frac{3 (2 x)}{5 x - 3}}{\frac{2 (0 + 3)}{5 x - 3}}$

Step 5.3.4.5.6

Add $0$ and $3$ .

$f (\frac{2 x}{5 x - 3}) = \frac{\frac{3 (2 x)}{5 x - 3}}{\frac{2 \cdot 3}{5 x - 3}}$

Step 5.3.4.6

Multiply $2$ by $3$ .

$f (\frac{2 x}{5 x - 3}) = \frac{\frac{3 (2 x)}{5 x - 3}}{\frac{6}{5 x - 3}}$

Step 5.3.5

Multiply $3$ by $2$ .

$f (\frac{2 x}{5 x - 3}) = \frac{\frac{6 x}{5 x - 3}}{\frac{6}{5 x - 3}}$

Step 5.3.6

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{2 x}{5 x - 3}) = \frac{6 x}{5 x - 3} \cdot \frac{5 x - 3}{6}$

Step 5.3.7

Cancel the common factor of $6$ .

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Step 5.3.7.1

Factor $6$ out of $6 x$ .

$f (\frac{2 x}{5 x - 3}) = \frac{6 (x)}{5 x - 3} \cdot \frac{5 x - 3}{6}$

Step 5.3.7.2

Cancel the common factor.

$f (\frac{2 x}{5 x - 3}) = \frac{6 x}{5 x - 3} \cdot \frac{5 x - 3}{6}$

Step 5.3.7.3

Rewrite the expression.

$f (\frac{2 x}{5 x - 3}) = \frac{x}{5 x - 3} \cdot (5 x - 3)$

Step 5.3.8

Cancel the common factor of $5 x - 3$ .

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Step 5.3.8.1

Cancel the common factor.

$f (\frac{2 x}{5 x - 3}) = \frac{x}{5 x - 3} \cdot (5 x - 3)$

Step 5.3.8.2

Rewrite the expression.

$f (\frac{2 x}{5 x - 3}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{2 x}{5 x - 3}$ is the inverse of $f (x) = \frac{3 x}{5 x - 2}$ .

$f^{- 1} (x) = \frac{2 x}{5 x - 3}$