Calculus Examples

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

Step 1

Write $f (x) = - \frac{2}{3} x + 6$ as an equation.

$y = - \frac{2}{3} x + 6$

Step 2

Interchange the variables.

$x = - \frac{2}{3} y + 6$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $- \frac{2}{3} y + 6 = x$ .

$- \frac{2}{3} y + 6 = x$

Step 3.2

Simplify each term.

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Step 3.2.1

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

$- \frac{y \cdot 2}{3} + 6 = x$

Step 3.2.2

Move $2$ to the left of $y$ .

$- \frac{2 y}{3} + 6 = x$

Step 3.3

Subtract $6$ from both sides of the equation.

$- \frac{2 y}{3} = x - 6$

Step 3.4

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

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

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 $- \frac{3}{2} (- \frac{2 y}{3})$ .

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Step 3.5.1.1.1

Cancel the common factor of $3$ .

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Step 3.5.1.1.1.1

Move the leading negative in $- \frac{3}{2}$ into the numerator.

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

Step 3.5.1.1.1.2

Move the leading negative in $- \frac{2 y}{3}$ into the numerator.

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

Step 3.5.1.1.1.3

Factor $3$ out of $- 3$ .

$\frac{3 (- 1)}{2} \cdot \frac{- 2 y}{3} = - \frac{3}{2} (x - 6)$

Step 3.5.1.1.1.4

Cancel the common factor.

$\frac{3 \cdot - 1}{2} \cdot \frac{- 2 y}{3} = - \frac{3}{2} (x - 6)$

Step 3.5.1.1.1.5

Rewrite the expression.

$\frac{- 1}{2} (- 2 y) = - \frac{3}{2} (x - 6)$

Step 3.5.1.1.2

Cancel the common factor of $2$ .

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Step 3.5.1.1.2.1

Factor $2$ out of $- 2 y$ .

$\frac{- 1}{2} (2 (- y)) = - \frac{3}{2} (x - 6)$

Step 3.5.1.1.2.2

Cancel the common factor.

$\frac{- 1}{2} (2 (- y)) = - \frac{3}{2} (x - 6)$

Step 3.5.1.1.2.3

Rewrite the expression.

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

Step 3.5.1.1.3

Multiply.

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Step 3.5.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 y = - \frac{3}{2} (x - 6)$

Step 3.5.1.1.3.2

Multiply $y$ by $1$ .

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

Step 3.5.2

Simplify the right side.

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Step 3.5.2.1

Simplify $- \frac{3}{2} (x - 6)$ .

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Step 3.5.2.1.1

Simplify terms.

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Step 3.5.2.1.1.1

Apply the distributive property.

$y = - \frac{3}{2} x - \frac{3}{2} \cdot - 6$

Step 3.5.2.1.1.2

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

$y = - \frac{x \cdot 3}{2} - \frac{3}{2} \cdot - 6$

Step 3.5.2.1.1.3

Cancel the common factor of $2$ .

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Step 3.5.2.1.1.3.1

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$y = - \frac{x \cdot 3}{2} + \frac{- 3}{2} \cdot - 6$

Step 3.5.2.1.1.3.2

Factor $2$ out of $- 6$ .

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

Step 3.5.2.1.1.3.3

Cancel the common factor.

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

Step 3.5.2.1.1.3.4

Rewrite the expression.

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

Step 3.5.2.1.1.4

Multiply $- 3$ by $- 3$ .

$y = - \frac{x \cdot 3}{2} + 9$

Step 3.5.2.1.2

Move $3$ to the left of $x$ .

$y = - \frac{3 x}{2} + 9$

Step 4

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

$f^{- 1} (x) = - \frac{3 x}{2} + 9$

Step 5

Verify if $f^{- 1} (x) = - \frac{3 x}{2} + 9$ is the inverse of $f (x) = - \frac{2}{3} x + 6$ .

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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 $f^{- 1} (- \frac{2}{3} x + 6)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (- \frac{2}{3} x + 6) = - \frac{3 (- \frac{2}{3} x + 6)}{2} + 9$

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

$f^{- 1} (- \frac{2}{3} x + 6) = - \frac{3 (- \frac{x \cdot 2}{3} + 6)}{2} + 9$

Step 5.2.3.1.2

Move $2$ to the left of $x$ .

$f^{- 1} (- \frac{2}{3} x + 6) = - \frac{3 (- \frac{2 \cdot x}{3} + 6)}{2} + 9$

Step 5.2.3.1.3

Factor $2$ out of $- \frac{2 x}{3} + 6$ .

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Step 5.2.3.1.3.1

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

$f^{- 1} (- \frac{2}{3} x + 6) = - \frac{3 (2 (- \frac{x}{3}) + 6)}{2} + 9$

Step 5.2.3.1.3.2

Factor $2$ out of $6$ .

$f^{- 1} (- \frac{2}{3} x + 6) = - \frac{3 (2 (- \frac{x}{3}) + 2 (3))}{2} + 9$

Step 5.2.3.1.3.3

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

$f^{- 1} (- \frac{2}{3} x + 6) = - \frac{3 (2 (- \frac{x}{3} + 3))}{2} + 9$

$f^{- 1} (- \frac{2}{3} x + 6) = - \frac{3 \cdot (2 (- \frac{x}{3} + 3))}{2} + 9$

Step 5.2.3.1.4

Multiply $3$ by $2$ .

$f^{- 1} (- \frac{2}{3} x + 6) = - \frac{6 (- \frac{x}{3} + 3)}{2} + 9$

Step 5.2.3.1.5

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

$f^{- 1} (- \frac{2}{3} x + 6) = - \frac{6 (- \frac{x}{3} + 3 \cdot \frac{3}{3})}{2} + 9$

Step 5.2.3.1.6

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

$f^{- 1} (- \frac{2}{3} x + 6) = - \frac{6 (- \frac{x}{3} + \frac{3 \cdot 3}{3})}{2} + 9$

Step 5.2.3.1.7

Combine the numerators over the common denominator.

$f^{- 1} (- \frac{2}{3} x + 6) = - \frac{6 (\frac{- x + 3 \cdot 3}{3})}{2} + 9$

Step 5.2.3.1.8

Multiply $3$ by $3$ .

$f^{- 1} (- \frac{2}{3} x + 6) = - \frac{6 (\frac{- x + 9}{3})}{2} + 9$

Step 5.2.3.2

Combine $6$ and $\frac{- x + 9}{3}$ .

$f^{- 1} (- \frac{2}{3} x + 6) = - \frac{\frac{6 (- x + 9)}{3}}{2} + 9$

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 $\frac{6 (- x + 9)}{3}$ by cancelling the common factors.

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Step 5.2.3.3.1.1

Factor $3$ out of $6 (- x + 9)$ .

$f^{- 1} (- \frac{2}{3} x + 6) = - \frac{\frac{3 (2 (- x + 9))}{3}}{2} + 9$

Step 5.2.3.3.1.2

Factor $3$ out of $3$ .

$f^{- 1} (- \frac{2}{3} x + 6) = - \frac{\frac{3 (2 (- x + 9))}{3 (1)}}{2} + 9$

Step 5.2.3.3.1.3

Cancel the common factor.

$f^{- 1} (- \frac{2}{3} x + 6) = - \frac{\frac{3 (2 (- x + 9))}{3 \cdot 1}}{2} + 9$

Step 5.2.3.3.1.4

Rewrite the expression.

$f^{- 1} (- \frac{2}{3} x + 6) = - \frac{\frac{2 (- x + 9)}{1}}{2} + 9$

Step 5.2.3.3.2

Divide $2 (- x + 9)$ by $1$ .

$f^{- 1} (- \frac{2}{3} x + 6) = - \frac{2 (- x + 9)}{2} + 9$

Step 5.2.3.4

Cancel the common factor of $2$ .

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Step 5.2.3.4.1

Cancel the common factor.

$f^{- 1} (- \frac{2}{3} x + 6) = - \frac{2 (- x + 9)}{2} + 9$

Step 5.2.3.4.2

Divide $- x + 9$ by $1$ .

$f^{- 1} (- \frac{2}{3} x + 6) = - (- x + 9) + 9$

Step 5.2.3.5

Apply the distributive property.

$f^{- 1} (- \frac{2}{3} x + 6) = x - 1 \cdot 9 + 9$

Step 5.2.3.6

Multiply $- - x$ .

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Step 5.2.3.6.1

Multiply $- 1$ by $- 1$ .

$f^{- 1} (- \frac{2}{3} x + 6) = 1 x - 1 \cdot 9 + 9$

Step 5.2.3.6.2

Multiply $x$ by $1$ .

$f^{- 1} (- \frac{2}{3} x + 6) = x - 1 \cdot 9 + 9$

Step 5.2.3.7

Multiply $- 1$ by $9$ .

$f^{- 1} (- \frac{2}{3} x + 6) = x - 9 + 9$

Step 5.2.4

Combine the opposite terms in $x - 9 + 9$ .

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Step 5.2.4.1

Add $- 9$ and $9$ .

$f^{- 1} (- \frac{2}{3} x + 6) = x + 0$

Step 5.2.4.2

Add $x$ and $0$ .

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

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 $f (- \frac{3 x}{2} + 9)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- \frac{3 x}{2} + 9) = - \frac{2}{3} \cdot (- \frac{3 x}{2} + 9) + 6$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Apply the distributive property.

$f (- \frac{3 x}{2} + 9) = - \frac{2}{3} \cdot (- \frac{3 x}{2}) - \frac{2}{3} \cdot 9 + 6$

Step 5.3.3.2

Cancel the common factor of $2$ .

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Step 5.3.3.2.1

Move the leading negative in $- \frac{2}{3}$ into the numerator.

$f (- \frac{3 x}{2} + 9) = \frac{- 2}{3} \cdot (- \frac{3 x}{2}) - \frac{2}{3} \cdot 9 + 6$

Step 5.3.3.2.2

Move the leading negative in $- \frac{3 x}{2}$ into the numerator.

$f (- \frac{3 x}{2} + 9) = \frac{- 2}{3} \cdot \frac{- 3 x}{2} - \frac{2}{3} \cdot 9 + 6$

Step 5.3.3.2.3

Factor $2$ out of $- 2$ .

$f (- \frac{3 x}{2} + 9) = \frac{2 (- 1)}{3} \cdot \frac{- 3 x}{2} - \frac{2}{3} \cdot 9 + 6$

Step 5.3.3.2.4

Cancel the common factor.

$f (- \frac{3 x}{2} + 9) = \frac{2 \cdot - 1}{3} \cdot \frac{- 3 x}{2} - \frac{2}{3} \cdot 9 + 6$

Step 5.3.3.2.5

Rewrite the expression.

$f (- \frac{3 x}{2} + 9) = \frac{- 1}{3} \cdot (- 3 x) - \frac{2}{3} \cdot 9 + 6$

Step 5.3.3.3

Cancel the common factor of $3$ .

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Step 5.3.3.3.1

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

$f (- \frac{3 x}{2} + 9) = \frac{- 1}{3} \cdot (3 (- x)) - \frac{2}{3} \cdot 9 + 6$

Step 5.3.3.3.2

Cancel the common factor.

$f (- \frac{3 x}{2} + 9) = \frac{- 1}{3} \cdot (3 (- x)) - \frac{2}{3} \cdot 9 + 6$

Step 5.3.3.3.3

Rewrite the expression.

$f (- \frac{3 x}{2} + 9) = - 1 (- x) - \frac{2}{3} \cdot 9 + 6$

Step 5.3.3.4

Multiply $- 1$ by $- 1$ .

$f (- \frac{3 x}{2} + 9) = 1 x - \frac{2}{3} \cdot 9 + 6$

Step 5.3.3.5

Multiply $x$ by $1$ .

$f (- \frac{3 x}{2} + 9) = x - \frac{2}{3} \cdot 9 + 6$

Step 5.3.3.6

Cancel the common factor of $3$ .

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Step 5.3.3.6.1

Move the leading negative in $- \frac{2}{3}$ into the numerator.

$f (- \frac{3 x}{2} + 9) = x + \frac{- 2}{3} \cdot 9 + 6$

Step 5.3.3.6.2

Factor $3$ out of $9$ .

$f (- \frac{3 x}{2} + 9) = x + \frac{- 2}{3} \cdot (3 (3)) + 6$

Step 5.3.3.6.3

Cancel the common factor.

$f (- \frac{3 x}{2} + 9) = x + \frac{- 2}{3} \cdot (3 \cdot 3) + 6$

Step 5.3.3.6.4

Rewrite the expression.

$f (- \frac{3 x}{2} + 9) = x - 2 \cdot 3 + 6$

Step 5.3.3.7

Multiply $- 2$ by $3$ .

$f (- \frac{3 x}{2} + 9) = x - 6 + 6$

Step 5.3.4

Combine the opposite terms in $x - 6 + 6$ .

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Step 5.3.4.1

Add $- 6$ and $6$ .

$f (- \frac{3 x}{2} + 9) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

$f (- \frac{3 x}{2} + 9) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{3 x}{2} + 9$ is the inverse of $f (x) = - \frac{2}{3} x + 6$ .

$f^{- 1} (x) = - \frac{3 x}{2} + 9$