Trigonometry Examples

$f (x) = \frac{4 x}{3 x - 5}$

Step 1

Write $f (x) = \frac{4 x}{3 x - 5}$ as an equation.

$y = \frac{4 x}{3 x - 5}$

Step 2

Interchange the variables.

$x = \frac{4 y}{3 y - 5}$

Step 3

Solve for $y$ .

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Step 3.1

Multiply the equation by $3 y - 5$ .

$(3 y - 5) (x) = (\frac{4 y}{3 y - 5}) (3 y - 5)$

Step 3.2

Simplify the left side.

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Step 3.2.1

Apply the distributive property.

$3 y x - 5 x = (\frac{4 y}{3 y - 5}) (3 y - 5)$

Step 3.3

Simplify the right side.

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Step 3.3.1

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

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Step 3.3.1.1

Cancel the common factor.

$3 y x - 5 x = \frac{4 y}{3 y - 5} (3 y - 5)$

Step 3.3.1.2

Rewrite the expression.

$3 y x - 5 x = 4 y$

Step 3.4

Solve for $y$ .

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Step 3.4.1

Subtract $4 y$ from both sides of the equation.

$3 y x - 5 x - 4 y = 0$

Step 3.4.2

Add $5 x$ to both sides of the equation.

$3 y x - 4 y = 5 x$

Step 3.4.3

Factor $y$ out of $3 y x - 4 y$ .

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Step 3.4.3.1

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

$y (3 x) - 4 y = 5 x$

Step 3.4.3.2

Factor $y$ out of $- 4 y$ .

$y (3 x) + y \cdot - 4 = 5 x$

Step 3.4.3.3

Factor $y$ out of $y (3 x) + y \cdot - 4$ .

$y (3 x - 4) = 5 x$

Step 3.4.4

Divide each term in $y (3 x - 4) = 5 x$ by $3 x - 4$ and simplify.

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Step 3.4.4.1

Divide each term in $y (3 x - 4) = 5 x$ by $3 x - 4$ .

$\frac{y (3 x - 4)}{3 x - 4} = \frac{5 x}{3 x - 4}$

Step 3.4.4.2

Simplify the left side.

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Step 3.4.4.2.1

Cancel the common factor of $3 x - 4$ .

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Step 3.4.4.2.1.1

Cancel the common factor.

$\frac{y (3 x - 4)}{3 x - 4} = \frac{5 x}{3 x - 4}$

Step 3.4.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{5 x}{3 x - 4}$

Step 4

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

$f^{- 1} (x) = \frac{5 x}{3 x - 4}$

Step 5

Verify if $f^{- 1} (x) = \frac{5 x}{3 x - 4}$ is the inverse of $f (x) = \frac{4 x}{3 x - 5}$ .

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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 $f^{- 1} (\frac{4 x}{3 x - 5})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{4 x}{3 x - 5}) = \frac{5 (\frac{4 x}{3 x - 5})}{3 (\frac{4 x}{3 x - 5}) - 4}$

Step 5.2.3

Combine $5$ and $\frac{4 x}{3 x - 5}$ .

$f^{- 1} (\frac{4 x}{3 x - 5}) = \frac{\frac{5 (4 x)}{3 x - 5}}{3 (\frac{4 x}{3 x - 5}) - 4}$

Step 5.2.4

Simplify the denominator.

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Step 5.2.4.1

Multiply $3 \frac{4 x}{3 x - 5}$ .

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Step 5.2.4.1.1

Combine $3$ and $\frac{4 x}{3 x - 5}$ .

$f^{- 1} (\frac{4 x}{3 x - 5}) = \frac{\frac{5 (4 x)}{3 x - 5}}{\frac{3 (4 x)}{3 x - 5} - 4}$

Step 5.2.4.1.2

Multiply $4$ by $3$ .

$f^{- 1} (\frac{4 x}{3 x - 5}) = \frac{\frac{5 (4 x)}{3 x - 5}}{\frac{12 x}{3 x - 5} - 4}$

Step 5.2.4.2

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{3 x - 5}{3 x - 5}$ .

$f^{- 1} (\frac{4 x}{3 x - 5}) = \frac{\frac{5 (4 x)}{3 x - 5}}{\frac{12 x}{3 x - 5} - 4 \cdot \frac{3 x - 5}{3 x - 5}}$

Step 5.2.4.3

Combine $- 4$ and $\frac{3 x - 5}{3 x - 5}$ .

$f^{- 1} (\frac{4 x}{3 x - 5}) = \frac{\frac{5 (4 x)}{3 x - 5}}{\frac{12 x}{3 x - 5} + \frac{- 4 (3 x - 5)}{3 x - 5}}$

Step 5.2.4.4

Combine the numerators over the common denominator.

$f^{- 1} (\frac{4 x}{3 x - 5}) = \frac{\frac{5 (4 x)}{3 x - 5}}{\frac{12 x - 4 (3 x - 5)}{3 x - 5}}$

Step 5.2.4.5

Rewrite $\frac{12 x - 4 (3 x - 5)}{3 x - 5}$ in a factored form.

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Step 5.2.4.5.1

Factor $4$ out of $12 x - 4 (3 x - 5)$ .

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Step 5.2.4.5.1.1

Factor $4$ out of $12 x$ .

$f^{- 1} (\frac{4 x}{3 x - 5}) = \frac{\frac{5 (4 x)}{3 x - 5}}{\frac{4 (3 x) - 4 (3 x - 5)}{3 x - 5}}$

Step 5.2.4.5.1.2

Factor $4$ out of $- 4 (3 x - 5)$ .

$f^{- 1} (\frac{4 x}{3 x - 5}) = \frac{\frac{5 (4 x)}{3 x - 5}}{\frac{4 (3 x) + 4 (- (3 x - 5))}{3 x - 5}}$

Step 5.2.4.5.1.3

Factor $4$ out of $4 (3 x) + 4 (- (3 x - 5))$ .

$f^{- 1} (\frac{4 x}{3 x - 5}) = \frac{\frac{5 (4 x)}{3 x - 5}}{\frac{4 (3 x - (3 x - 5))}{3 x - 5}}$

Step 5.2.4.5.2

Apply the distributive property.

$f^{- 1} (\frac{4 x}{3 x - 5}) = \frac{\frac{5 (4 x)}{3 x - 5}}{\frac{4 (3 x - (3 x) + 5)}{3 x - 5}}$

Step 5.2.4.5.3

Multiply $3$ by $- 1$ .

$f^{- 1} (\frac{4 x}{3 x - 5}) = \frac{\frac{5 (4 x)}{3 x - 5}}{\frac{4 (3 x - 3 x + 5)}{3 x - 5}}$

Step 5.2.4.5.4

Multiply $- 1$ by $- 5$ .

$f^{- 1} (\frac{4 x}{3 x - 5}) = \frac{\frac{5 (4 x)}{3 x - 5}}{\frac{4 (3 x - 3 x + 5)}{3 x - 5}}$

Step 5.2.4.5.5

Subtract $3 x$ from $3 x$ .

$f^{- 1} (\frac{4 x}{3 x - 5}) = \frac{\frac{5 (4 x)}{3 x - 5}}{\frac{4 (0 + 5)}{3 x - 5}}$

Step 5.2.4.5.6

Add $0$ and $5$ .

$f^{- 1} (\frac{4 x}{3 x - 5}) = \frac{\frac{5 (4 x)}{3 x - 5}}{\frac{4 \cdot 5}{3 x - 5}}$

Step 5.2.4.6

Multiply $4$ by $5$ .

$f^{- 1} (\frac{4 x}{3 x - 5}) = \frac{\frac{5 (4 x)}{3 x - 5}}{\frac{20}{3 x - 5}}$

Step 5.2.5

Multiply $5$ by $4$ .

$f^{- 1} (\frac{4 x}{3 x - 5}) = \frac{\frac{20 x}{3 x - 5}}{\frac{20}{3 x - 5}}$

Step 5.2.6

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{4 x}{3 x - 5}) = \frac{20 x}{3 x - 5} \cdot \frac{3 x - 5}{20}$

Step 5.2.7

Cancel the common factor of $20$ .

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Step 5.2.7.1

Factor $20$ out of $20 x$ .

$f^{- 1} (\frac{4 x}{3 x - 5}) = \frac{20 (x)}{3 x - 5} \cdot \frac{3 x - 5}{20}$

Step 5.2.7.2

Cancel the common factor.

$f^{- 1} (\frac{4 x}{3 x - 5}) = \frac{20 x}{3 x - 5} \cdot \frac{3 x - 5}{20}$

Step 5.2.7.3

Rewrite the expression.

$f^{- 1} (\frac{4 x}{3 x - 5}) = \frac{x}{3 x - 5} \cdot (3 x - 5)$

Step 5.2.8

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

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Step 5.2.8.1

Cancel the common factor.

$f^{- 1} (\frac{4 x}{3 x - 5}) = \frac{x}{3 x - 5} \cdot (3 x - 5)$

Step 5.2.8.2

Rewrite the expression.

$f^{- 1} (\frac{4 x}{3 x - 5}) = x$

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 $f (\frac{5 x}{3 x - 4})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{5 x}{3 x - 4}) = \frac{4 (\frac{5 x}{3 x - 4})}{3 (\frac{5 x}{3 x - 4}) - 5}$

Step 5.3.3

Combine $4$ and $\frac{5 x}{3 x - 4}$ .

$f (\frac{5 x}{3 x - 4}) = \frac{\frac{4 (5 x)}{3 x - 4}}{3 (\frac{5 x}{3 x - 4}) - 5}$

Step 5.3.4

Simplify the denominator.

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Step 5.3.4.1

Multiply $3 \frac{5 x}{3 x - 4}$ .

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Step 5.3.4.1.1

Combine $3$ and $\frac{5 x}{3 x - 4}$ .

$f (\frac{5 x}{3 x - 4}) = \frac{\frac{4 (5 x)}{3 x - 4}}{\frac{3 (5 x)}{3 x - 4} - 5}$

Step 5.3.4.1.2

Multiply $5$ by $3$ .

$f (\frac{5 x}{3 x - 4}) = \frac{\frac{4 (5 x)}{3 x - 4}}{\frac{15 x}{3 x - 4} - 5}$

Step 5.3.4.2

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{3 x - 4}{3 x - 4}$ .

$f (\frac{5 x}{3 x - 4}) = \frac{\frac{4 (5 x)}{3 x - 4}}{\frac{15 x}{3 x - 4} - 5 \cdot \frac{3 x - 4}{3 x - 4}}$

Step 5.3.4.3

Combine $- 5$ and $\frac{3 x - 4}{3 x - 4}$ .

$f (\frac{5 x}{3 x - 4}) = \frac{\frac{4 (5 x)}{3 x - 4}}{\frac{15 x}{3 x - 4} + \frac{- 5 (3 x - 4)}{3 x - 4}}$

Step 5.3.4.4

Combine the numerators over the common denominator.

$f (\frac{5 x}{3 x - 4}) = \frac{\frac{4 (5 x)}{3 x - 4}}{\frac{15 x - 5 (3 x - 4)}{3 x - 4}}$

Step 5.3.4.5

Rewrite $\frac{15 x - 5 (3 x - 4)}{3 x - 4}$ in a factored form.

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Step 5.3.4.5.1

Factor $5$ out of $15 x - 5 (3 x - 4)$ .

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Step 5.3.4.5.1.1

Factor $5$ out of $15 x$ .

$f (\frac{5 x}{3 x - 4}) = \frac{\frac{4 (5 x)}{3 x - 4}}{\frac{5 (3 x) - 5 (3 x - 4)}{3 x - 4}}$

Step 5.3.4.5.1.2

Factor $5$ out of $- 5 (3 x - 4)$ .

$f (\frac{5 x}{3 x - 4}) = \frac{\frac{4 (5 x)}{3 x - 4}}{\frac{5 (3 x) + 5 (- (3 x - 4))}{3 x - 4}}$

Step 5.3.4.5.1.3

Factor $5$ out of $5 (3 x) + 5 (- (3 x - 4))$ .

$f (\frac{5 x}{3 x - 4}) = \frac{\frac{4 (5 x)}{3 x - 4}}{\frac{5 (3 x - (3 x - 4))}{3 x - 4}}$

Step 5.3.4.5.2

Apply the distributive property.

$f (\frac{5 x}{3 x - 4}) = \frac{\frac{4 (5 x)}{3 x - 4}}{\frac{5 (3 x - (3 x) + 4)}{3 x - 4}}$

Step 5.3.4.5.3

Multiply $3$ by $- 1$ .

$f (\frac{5 x}{3 x - 4}) = \frac{\frac{4 (5 x)}{3 x - 4}}{\frac{5 (3 x - 3 x + 4)}{3 x - 4}}$

Step 5.3.4.5.4

Multiply $- 1$ by $- 4$ .

$f (\frac{5 x}{3 x - 4}) = \frac{\frac{4 (5 x)}{3 x - 4}}{\frac{5 (3 x - 3 x + 4)}{3 x - 4}}$

Step 5.3.4.5.5

Subtract $3 x$ from $3 x$ .

$f (\frac{5 x}{3 x - 4}) = \frac{\frac{4 (5 x)}{3 x - 4}}{\frac{5 (0 + 4)}{3 x - 4}}$

Step 5.3.4.5.6

Add $0$ and $4$ .

$f (\frac{5 x}{3 x - 4}) = \frac{\frac{4 (5 x)}{3 x - 4}}{\frac{5 \cdot 4}{3 x - 4}}$

Step 5.3.4.6

Multiply $5$ by $4$ .

$f (\frac{5 x}{3 x - 4}) = \frac{\frac{4 (5 x)}{3 x - 4}}{\frac{20}{3 x - 4}}$

Step 5.3.5

Multiply $4$ by $5$ .

$f (\frac{5 x}{3 x - 4}) = \frac{\frac{20 x}{3 x - 4}}{\frac{20}{3 x - 4}}$

Step 5.3.6

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{5 x}{3 x - 4}) = \frac{20 x}{3 x - 4} \cdot \frac{3 x - 4}{20}$

Step 5.3.7

Cancel the common factor of $20$ .

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Step 5.3.7.1

Factor $20$ out of $20 x$ .

$f (\frac{5 x}{3 x - 4}) = \frac{20 (x)}{3 x - 4} \cdot \frac{3 x - 4}{20}$

Step 5.3.7.2

Cancel the common factor.

$f (\frac{5 x}{3 x - 4}) = \frac{20 x}{3 x - 4} \cdot \frac{3 x - 4}{20}$

Step 5.3.7.3

Rewrite the expression.

$f (\frac{5 x}{3 x - 4}) = \frac{x}{3 x - 4} \cdot (3 x - 4)$

Step 5.3.8

Cancel the common factor of $3 x - 4$ .

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Step 5.3.8.1

Cancel the common factor.

$f (\frac{5 x}{3 x - 4}) = \frac{x}{3 x - 4} \cdot (3 x - 4)$

Step 5.3.8.2

Rewrite the expression.

$f (\frac{5 x}{3 x - 4}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{5 x}{3 x - 4}$ is the inverse of $f (x) = \frac{4 x}{3 x - 5}$ .

$f^{- 1} (x) = \frac{5 x}{3 x - 4}$