Algebra Examples

$f (x) = \frac{3 x - 15}{2}$

Step 1

Write $f (x) = \frac{3 x - 15}{2}$ as an equation.

$y = \frac{3 x - 15}{2}$

Step 2

Interchange the variables.

$x = \frac{3 y - 15}{2}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{3 y - 15}{2} = x$ .

$\frac{3 y - 15}{2} = x$

Step 3.2

Multiply both sides by $2$ .

$\frac{3 y - 15}{2} \cdot 2 = x \cdot 2$

Step 3.3

Simplify.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

Simplify $\frac{3 y - 15}{2} \cdot 2$ .

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Step 3.3.1.1.1

Factor $3$ out of $3 y - 15$ .

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Step 3.3.1.1.1.1

Factor $3$ out of $3 y$ .

$\frac{3 (y) - 15}{2} \cdot 2 = x \cdot 2$

Step 3.3.1.1.1.2

Factor $3$ out of $- 15$ .

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

Step 3.3.1.1.1.3

Factor $3$ out of $3 y + 3 \cdot - 5$ .

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

Step 3.3.1.1.2

Cancel the common factor of $2$ .

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Step 3.3.1.1.2.1

Cancel the common factor.

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

Step 3.3.1.1.2.2

Rewrite the expression.

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

Step 3.3.1.1.3

Apply the distributive property.

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

Step 3.3.1.1.4

Multiply $3$ by $- 5$ .

$3 y - 15 = x \cdot 2$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

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

$3 y - 15 = 2 x$

Step 3.4

Solve for $y$ .

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Step 3.4.1

Add $15$ to both sides of the equation.

$3 y = 2 x + 15$

Step 3.4.2

Divide each term in $3 y = 2 x + 15$ by $3$ and simplify.

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Step 3.4.2.1

Divide each term in $3 y = 2 x + 15$ by $3$ .

$\frac{3 y}{3} = \frac{2 x}{3} + \frac{15}{3}$

Step 3.4.2.2

Simplify the left side.

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Step 3.4.2.2.1

Cancel the common factor of $3$ .

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Step 3.4.2.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{2 x}{3} + \frac{15}{3}$

Step 3.4.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{2 x}{3} + \frac{15}{3}$

Step 3.4.2.3

Simplify the right side.

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Step 3.4.2.3.1

Divide $15$ by $3$ .

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

Step 4

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

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

Step 5

Verify if $f^{- 1} (x) = \frac{2 x}{3} + 5$ is the inverse of $f (x) = \frac{3 x - 15}{2}$ .

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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 $f^{- 1} (\frac{3 x - 15}{2})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{3 x - 15}{2}) = \frac{2 (\frac{3 x - 15}{2})}{3} + 5$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Factor $3$ out of $3 x - 15$ .

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Step 5.2.3.1.1

Factor $3$ out of $3 x$ .

$f^{- 1} (\frac{3 x - 15}{2}) = \frac{2 (\frac{3 (x) - 15}{2})}{3} + 5$

Step 5.2.3.1.2

Factor $3$ out of $- 15$ .

$f^{- 1} (\frac{3 x - 15}{2}) = \frac{2 (\frac{3 x + 3 \cdot - 5}{2})}{3} + 5$

Step 5.2.3.1.3

Factor $3$ out of $3 x + 3 \cdot - 5$ .

$f^{- 1} (\frac{3 x - 15}{2}) = \frac{2 (\frac{3 (x - 5)}{2})}{3} + 5$

Step 5.2.3.2

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

$f^{- 1} (\frac{3 x - 15}{2}) = \frac{\frac{2 (3 (x - 5))}{2}}{3} + 5$

Step 5.2.3.3

Multiply $2$ by $3$ .

$f^{- 1} (\frac{3 x - 15}{2}) = \frac{\frac{6 (x - 5)}{2}}{3} + 5$

Step 5.2.3.4

Reduce the expression by cancelling the common factors.

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Step 5.2.3.4.1

Reduce the expression $\frac{6 (x - 5)}{2}$ by cancelling the common factors.

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Step 5.2.3.4.1.1

Factor $2$ out of $6 (x - 5)$ .

$f^{- 1} (\frac{3 x - 15}{2}) = \frac{\frac{2 (3 (x - 5))}{2}}{3} + 5$

Step 5.2.3.4.1.2

Factor $2$ out of $2$ .

$f^{- 1} (\frac{3 x - 15}{2}) = \frac{\frac{2 (3 (x - 5))}{2 (1)}}{3} + 5$

Step 5.2.3.4.1.3

Cancel the common factor.

$f^{- 1} (\frac{3 x - 15}{2}) = \frac{\frac{2 (3 (x - 5))}{2 \cdot 1}}{3} + 5$

Step 5.2.3.4.1.4

Rewrite the expression.

$f^{- 1} (\frac{3 x - 15}{2}) = \frac{\frac{3 (x - 5)}{1}}{3} + 5$

Step 5.2.3.4.2

Divide $3 (x - 5)$ by $1$ .

$f^{- 1} (\frac{3 x - 15}{2}) = \frac{3 (x - 5)}{3} + 5$

Step 5.2.3.5

Cancel the common factor of $3$ .

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Step 5.2.3.5.1

Cancel the common factor.

$f^{- 1} (\frac{3 x - 15}{2}) = \frac{3 (x - 5)}{3} + 5$

Step 5.2.3.5.2

Divide $x - 5$ by $1$ .

$f^{- 1} (\frac{3 x - 15}{2}) = x - 5 + 5$

Step 5.2.4

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

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Step 5.2.4.1

Add $- 5$ and $5$ .

$f^{- 1} (\frac{3 x - 15}{2}) = x + 0$

Step 5.2.4.2

Add $x$ and $0$ .

$f^{- 1} (\frac{3 x - 15}{2}) = x$

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

$f (\frac{2 x}{3} + 5) = \frac{3 (\frac{2 x}{3} + 5) - 15}{2}$

Step 5.3.3

Simplify the numerator.

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Step 5.3.3.1

Factor $3$ out of $3 (\frac{2 x}{3} + 5) - 15$ .

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Step 5.3.3.1.1

Factor $3$ out of $- 15$ .

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

Step 5.3.3.1.2

Factor $3$ out of $3 (\frac{2 x}{3} + 5) + 3 \cdot - 5$ .

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

Step 5.3.3.2

Subtract $5$ from $5$ .

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

Step 5.3.3.3

Add $\frac{2 x}{3}$ and $0$ .

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

Step 5.3.4

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

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

Step 5.3.5

Multiply $3$ by $2$ .

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

Step 5.3.6

Reduce the expression by cancelling the common factors.

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Step 5.3.6.1

Reduce the expression $\frac{6 x}{3}$ by cancelling the common factors.

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Step 5.3.6.1.1

Factor $3$ out of $6 x$ .

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

Step 5.3.6.1.2

Factor $3$ out of $3$ .

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

Step 5.3.6.1.3

Cancel the common factor.

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

Step 5.3.6.1.4

Rewrite the expression.

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

Step 5.3.6.2

Divide $2 x$ by $1$ .

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

Step 5.3.7

Cancel the common factor of $2$ .

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Step 5.3.7.1

Cancel the common factor.

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

Step 5.3.7.2

Divide $x$ by $1$ .

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

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{2 x}{3} + 5$ is the inverse of $f (x) = \frac{3 x - 15}{2}$ .

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