Calculus Examples

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

Step 1

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

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{2}{3 y - 5} = x$ .

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

Step 3.2

Find the LCD of the terms in the equation.

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Step 3.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$3 y - 5, 1$

Step 3.2.2

Remove parentheses.

$3 y - 5, 1$

Step 3.2.3

The LCM of one and any expression is the expression.

$3 y - 5$

Step 3.3

Multiply each term in $\frac{2}{3 y - 5} = x$ by $3 y - 5$ to eliminate the fractions.

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Step 3.3.1

Multiply each term in $\frac{2}{3 y - 5} = x$ by $3 y - 5$ .

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

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

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

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Step 3.3.2.1.1

Cancel the common factor.

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

Step 3.3.2.1.2

Rewrite the expression.

$2 = x (3 y - 5)$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Apply the distributive property.

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

Step 3.3.3.2

Reorder.

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Step 3.3.3.2.1

Rewrite using the commutative property of multiplication.

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

Step 3.3.3.2.2

Move $- 5$ to the left of $x$ .

$2 = 3 x y - 5 x$

Step 3.4

Solve the equation.

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Step 3.4.1

Rewrite the equation as $3 x y - 5 x = 2$ .

$3 x y - 5 x = 2$

Step 3.4.2

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

$3 x y = 2 + 5 x$

Step 3.4.3

Divide each term in $3 x y = 2 + 5 x$ by $3 x$ and simplify.

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Step 3.4.3.1

Divide each term in $3 x y = 2 + 5 x$ by $3 x$ .

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

Step 3.4.3.2

Simplify the left side.

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Step 3.4.3.2.1

Cancel the common factor of $3$ .

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Step 3.4.3.2.1.1

Cancel the common factor.

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

Step 3.4.3.2.1.2

Rewrite the expression.

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

Step 3.4.3.2.2

Cancel the common factor of $x$ .

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Step 3.4.3.2.2.1

Cancel the common factor.

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

Step 3.4.3.2.2.2

Divide $y$ by $1$ .

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

Step 3.4.3.3

Simplify the right side.

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Step 3.4.3.3.1

Cancel the common factor of $x$ .

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Step 3.4.3.3.1.1

Cancel the common factor.

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

Step 3.4.3.3.1.2

Rewrite the expression.

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

Step 4

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

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

Step 5

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

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

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

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

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

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

Step 5.2.3.2

Multiply $3$ by $2$ .

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

Step 5.2.3.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.2.3.4

Cancel the common factor of $2$ .

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Step 5.2.3.4.1

Factor $2$ out of $6$ .

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

Step 5.2.3.4.2

Cancel the common factor.

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

Step 5.2.3.4.3

Rewrite the expression.

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

Step 5.2.4

Simplify terms.

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Step 5.2.4.1

Combine the numerators over the common denominator.

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

Step 5.2.4.2

Combine the opposite terms in $3 x - 5 + 5$ .

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Step 5.2.4.2.1

Add $- 5$ and $5$ .

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

Step 5.2.4.2.2

Add $3 x$ and $0$ .

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

Step 5.2.4.3

Cancel the common factor of $3$ .

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Step 5.2.4.3.1

Cancel the common factor.

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

Step 5.2.4.3.2

Divide $x$ by $1$ .

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

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

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

Step 5.3.3

Simplify the denominator.

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Step 5.3.3.1

Apply the distributive property.

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

Step 5.3.3.2

Cancel the common factor of $3$ .

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Step 5.3.3.2.1

Factor $3$ out of $3 x$ .

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

Step 5.3.3.2.2

Cancel the common factor.

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

Step 5.3.3.2.3

Rewrite the expression.

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

Step 5.3.3.3

Cancel the common factor of $3$ .

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Step 5.3.3.3.1

Cancel the common factor.

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

Step 5.3.3.3.2

Rewrite the expression.

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

Step 5.3.3.4

Subtract $5$ from $5$ .

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

Step 5.3.3.5

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

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

Step 5.3.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.3.5

Cancel the common factor of $2$ .

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Step 5.3.5.1

Cancel the common factor.

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

Step 5.3.5.2

Rewrite the expression.

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

Step 5.4

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

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