Algebra Examples

$f (x) = \frac{2 x}{3} - 17$

Step 1

Write $f (x) = \frac{2 x}{3} - 17$ as an equation.

$y = \frac{2 x}{3} - 17$

Step 2

Interchange the variables.

$x = \frac{2 y}{3} - 17$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{2 y}{3} - 17 = x$ .

$\frac{2 y}{3} - 17 = x$

Step 3.2

Add $17$ to both sides of the equation.

$\frac{2 y}{3} = x + 17$

Step 3.3

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

$\frac{3}{2} \cdot \frac{2 y}{3} = \frac{3}{2} (x + 17)$

Step 3.4

Simplify both sides of the equation.

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Step 3.4.1

Simplify the left side.

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Step 3.4.1.1

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

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Step 3.4.1.1.1

Cancel the common factor of $3$ .

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Step 3.4.1.1.1.1

Cancel the common factor.

$\frac{3}{2} \cdot \frac{2 y}{3} = \frac{3}{2} (x + 17)$

Step 3.4.1.1.1.2

Rewrite the expression.

$\frac{1}{2} (2 y) = \frac{3}{2} (x + 17)$

Step 3.4.1.1.2

Cancel the common factor of $2$ .

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Step 3.4.1.1.2.1

Factor $2$ out of $2 y$ .

$\frac{1}{2} (2 (y)) = \frac{3}{2} (x + 17)$

Step 3.4.1.1.2.2

Cancel the common factor.

$\frac{1}{2} (2 y) = \frac{3}{2} (x + 17)$

Step 3.4.1.1.2.3

Rewrite the expression.

$y = \frac{3}{2} (x + 17)$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $\frac{3}{2} (x + 17)$ .

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Step 3.4.2.1.1

Apply the distributive property.

$y = \frac{3}{2} x + \frac{3}{2} \cdot 17$

Step 3.4.2.1.2

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

$y = \frac{3 x}{2} + \frac{3}{2} \cdot 17$

Step 3.4.2.1.3

Multiply $\frac{3}{2} \cdot 17$ .

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Step 3.4.2.1.3.1

Combine $\frac{3}{2}$ and $17$ .

$y = \frac{3 x}{2} + \frac{3 \cdot 17}{2}$

Step 3.4.2.1.3.2

Multiply $3$ by $17$ .

$y = \frac{3 x}{2} + \frac{51}{2}$

Step 4

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

$f^{- 1} (x) = \frac{3 x}{2} + \frac{51}{2}$

Step 5

Verify if $f^{- 1} (x) = \frac{3 x}{2} + \frac{51}{2}$ is the inverse of $f (x) = \frac{2 x}{3} - 17$ .

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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 $f^{- 1} (\frac{2 x}{3} - 17)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{2 x}{3} - 17) = \frac{3 (\frac{2 x}{3} - 17)}{2} + \frac{51}{2}$

Step 5.2.3

Combine the numerators over the common denominator.

$f^{- 1} (\frac{2 x}{3} - 17) = \frac{3 (\frac{2 x}{3} - 17) + 51}{2}$

Step 5.2.4

Simplify each term.

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Step 5.2.4.1

Apply the distributive property.

$f^{- 1} (\frac{2 x}{3} - 17) = \frac{3 (\frac{2 x}{3}) + 3 \cdot - 17 + 51}{2}$

Step 5.2.4.2

Cancel the common factor of $3$ .

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Step 5.2.4.2.1

Cancel the common factor.

$f^{- 1} (\frac{2 x}{3} - 17) = \frac{3 (\frac{2 x}{3}) + 3 \cdot - 17 + 51}{2}$

Step 5.2.4.2.2

Rewrite the expression.

$f^{- 1} (\frac{2 x}{3} - 17) = \frac{2 x + 3 \cdot - 17 + 51}{2}$

Step 5.2.4.3

Multiply $3$ by $- 17$ .

$f^{- 1} (\frac{2 x}{3} - 17) = \frac{2 x - 51 + 51}{2}$

Step 5.2.5

Simplify terms.

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Step 5.2.5.1

Combine the opposite terms in $2 x - 51 + 51$ .

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Step 5.2.5.1.1

Add $- 51$ and $51$ .

$f^{- 1} (\frac{2 x}{3} - 17) = \frac{2 x + 0}{2}$

Step 5.2.5.1.2

Add $2 x$ and $0$ .

$f^{- 1} (\frac{2 x}{3} - 17) = \frac{2 x}{2}$

Step 5.2.5.2

Cancel the common factor of $2$ .

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Step 5.2.5.2.1

Cancel the common factor.

$f^{- 1} (\frac{2 x}{3} - 17) = \frac{2 x}{2}$

Step 5.2.5.2.2

Divide $x$ by $1$ .

$f^{- 1} (\frac{2 x}{3} - 17) = x$

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 $f (\frac{3 x}{2} + \frac{51}{2})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{3 x}{2} + \frac{51}{2}) = \frac{2 (\frac{3 x}{2} + \frac{51}{2})}{3} - 17$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Simplify the numerator.

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Step 5.3.3.1.1

Combine the numerators over the common denominator.

$f (\frac{3 x}{2} + \frac{51}{2}) = \frac{2 (\frac{3 x + 51}{2})}{3} - 17$

Step 5.3.3.1.2

Factor $3$ out of $3 x + 51$ .

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Step 5.3.3.1.2.1

Factor $3$ out of $3 x$ .

$f (\frac{3 x}{2} + \frac{51}{2}) = \frac{2 (\frac{3 (x) + 51}{2})}{3} - 17$

Step 5.3.3.1.2.2

Factor $3$ out of $51$ .

$f (\frac{3 x}{2} + \frac{51}{2}) = \frac{2 (\frac{3 x + 3 \cdot 17}{2})}{3} - 17$

Step 5.3.3.1.2.3

Factor $3$ out of $3 x + 3 \cdot 17$ .

$f (\frac{3 x}{2} + \frac{51}{2}) = \frac{2 (\frac{3 (x + 17)}{2})}{3} - 17$

Step 5.3.3.2

Combine $2$ and $\frac{3 (x + 17)}{2}$ .

$f (\frac{3 x}{2} + \frac{51}{2}) = \frac{\frac{2 (3 (x + 17))}{2}}{3} - 17$

Step 5.3.3.3

Multiply $2$ by $3$ .

$f (\frac{3 x}{2} + \frac{51}{2}) = \frac{\frac{6 (x + 17)}{2}}{3} - 17$

Step 5.3.3.4

Reduce the expression by cancelling the common factors.

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Step 5.3.3.4.1

Reduce the expression $\frac{6 (x + 17)}{2}$ by cancelling the common factors.

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Step 5.3.3.4.1.1

Factor $2$ out of $6 (x + 17)$ .

$f (\frac{3 x}{2} + \frac{51}{2}) = \frac{\frac{2 (3 (x + 17))}{2}}{3} - 17$

Step 5.3.3.4.1.2

Factor $2$ out of $2$ .

$f (\frac{3 x}{2} + \frac{51}{2}) = \frac{\frac{2 (3 (x + 17))}{2 (1)}}{3} - 17$

Step 5.3.3.4.1.3

Cancel the common factor.

$f (\frac{3 x}{2} + \frac{51}{2}) = \frac{\frac{2 (3 (x + 17))}{2 \cdot 1}}{3} - 17$

Step 5.3.3.4.1.4

Rewrite the expression.

$f (\frac{3 x}{2} + \frac{51}{2}) = \frac{\frac{3 (x + 17)}{1}}{3} - 17$

Step 5.3.3.4.2

Divide $3 (x + 17)$ by $1$ .

$f (\frac{3 x}{2} + \frac{51}{2}) = \frac{3 (x + 17)}{3} - 17$

Step 5.3.3.5

Cancel the common factor of $3$ .

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Step 5.3.3.5.1

Cancel the common factor.

$f (\frac{3 x}{2} + \frac{51}{2}) = \frac{3 (x + 17)}{3} - 17$

Step 5.3.3.5.2

Divide $x + 17$ by $1$ .

$f (\frac{3 x}{2} + \frac{51}{2}) = x + 17 - 17$

Step 5.3.4

Combine the opposite terms in $x + 17 - 17$ .

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Step 5.3.4.1

Subtract $17$ from $17$ .

$f (\frac{3 x}{2} + \frac{51}{2}) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

$f (\frac{3 x}{2} + \frac{51}{2}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{3 x}{2} + \frac{51}{2}$ is the inverse of $f (x) = \frac{2 x}{3} - 17$ .

$f^{- 1} (x) = \frac{3 x}{2} + \frac{51}{2}$