Calculus Examples

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

Step 1

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

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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Step 3.1

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

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

Step 3.2

Combine $\frac{3}{2}$ and $y$ .

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

Step 3.3

Subtract $3$ from both sides of the equation.

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

Step 3.4

Multiply both sides of the equation by $\frac{2}{3}$ .

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

Step 3.5

Simplify both sides of the equation.

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Step 3.5.1

Simplify the left side.

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Step 3.5.1.1

Simplify $\frac{2}{3} \cdot \frac{3 y}{2}$ .

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Step 3.5.1.1.1

Cancel the common factor of $2$ .

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Step 3.5.1.1.1.1

Cancel the common factor.

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

Step 3.5.1.1.1.2

Rewrite the expression.

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

Step 3.5.1.1.2

Cancel the common factor of $3$ .

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Step 3.5.1.1.2.1

Factor $3$ out of $3 y$ .

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

Step 3.5.1.1.2.2

Cancel the common factor.

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

Step 3.5.1.1.2.3

Rewrite the expression.

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

Step 3.5.2

Simplify the right side.

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Step 3.5.2.1

Simplify $\frac{2}{3} (x - 3)$ .

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Step 3.5.2.1.1

Apply the distributive property.

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

Step 3.5.2.1.2

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

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

Step 3.5.2.1.3

Cancel the common factor of $3$ .

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Step 3.5.2.1.3.1

Factor $3$ out of $- 3$ .

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

Step 3.5.2.1.3.2

Cancel the common factor.

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

Step 3.5.2.1.3.3

Rewrite the expression.

$y = \frac{2 x}{3} + 2 \cdot - 1$

Step 3.5.2.1.4

Multiply $2$ by $- 1$ .

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

Step 4

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

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

Step 5

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

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

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

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Simplify the numerator.

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Step 5.2.3.1.1

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

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

Step 5.2.3.1.2

Factor $3$ out of $\frac{3 x}{2} + 3$ .

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Step 5.2.3.1.2.1

Factor $3$ out of $\frac{3 x}{2}$ .

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

Step 5.2.3.1.2.2

Factor $3$ out of $3$ .

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

Step 5.2.3.1.2.3

Factor $3$ out of $3 \frac{x}{2} + 3 (1)$ .

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

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

Step 5.2.3.1.3

Multiply $2$ by $3$ .

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

Step 5.2.3.1.4

Write $1$ as a fraction with a common denominator.

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

Step 5.2.3.1.5

Combine the numerators over the common denominator.

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

Step 5.2.3.2

Combine $6$ and $\frac{x + 2}{2}$ .

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

Step 5.2.3.3

Reduce the expression by cancelling the common factors.

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Step 5.2.3.3.1

Reduce the expression $\frac{6 (x + 2)}{2}$ by cancelling the common factors.

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Step 5.2.3.3.1.1

Factor $2$ out of $6 (x + 2)$ .

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

Step 5.2.3.3.1.2

Factor $2$ out of $2$ .

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

Step 5.2.3.3.1.3

Cancel the common factor.

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

Step 5.2.3.3.1.4

Rewrite the expression.

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

Step 5.2.3.3.2

Divide $3 (x + 2)$ by $1$ .

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

Step 5.2.3.4

Cancel the common factor of $3$ .

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Step 5.2.3.4.1

Cancel the common factor.

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

Step 5.2.3.4.2

Divide $x + 2$ by $1$ .

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

Step 5.2.4

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

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Step 5.2.4.1

Subtract $2$ from $2$ .

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

Step 5.2.4.2

Add $x$ and $0$ .

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

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

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

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Apply the distributive property.

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

Step 5.3.3.2

Cancel the common factor of $3$ .

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Step 5.3.3.2.1

Cancel the common factor.

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

Step 5.3.3.2.2

Rewrite the expression.

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

Step 5.3.3.3

Cancel the common factor of $2$ .

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Step 5.3.3.3.1

Factor $2$ out of $2 x$ .

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

Step 5.3.3.3.2

Cancel the common factor.

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

Step 5.3.3.3.3

Rewrite the expression.

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

Step 5.3.3.4

Cancel the common factor of $2$ .

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Step 5.3.3.4.1

Factor $2$ out of $- 2$ .

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

Step 5.3.3.4.2

Cancel the common factor.

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

Step 5.3.3.4.3

Rewrite the expression.

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

Step 5.3.3.5

Multiply $3$ by $- 1$ .

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

Step 5.3.4

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

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Step 5.3.4.1

Add $- 3$ and $3$ .

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

Step 5.3.4.2

Add $x$ and $0$ .

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

Step 5.4

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

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