Algebra Examples

$f (x) = \frac{1}{2} (3 x + 4)$

Step 1

Write $f (x) = \frac{1}{2} (3 x + 4)$ as an equation.

$y = \frac{1}{2} (3 x + 4)$

Step 2

Interchange the variables.

$x = \frac{1}{2} \cdot (3 y + 4)$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{1}{2} \cdot (3 y + 4) = x$ .

$\frac{1}{2} \cdot (3 y + 4) = x$

Step 3.2

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} \cdot (3 y + 4)) = 2 x$

Step 3.3

Simplify the left side.

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Step 3.3.1

Simplify $2 (\frac{1}{2} \cdot (3 y + 4))$ .

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Step 3.3.1.1

Apply the distributive property.

$2 (\frac{1}{2} (3 y) + \frac{1}{2} \cdot 4) = 2 x$

Step 3.3.1.2

Multiply $\frac{1}{2} (3 y)$ .

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Step 3.3.1.2.1

Combine $3$ and $\frac{1}{2}$ .

$2 (\frac{3}{2} y + \frac{1}{2} \cdot 4) = 2 x$

Step 3.3.1.2.2

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

$2 (\frac{3 y}{2} + \frac{1}{2} \cdot 4) = 2 x$

Step 3.3.1.3

Cancel the common factor of $2$ .

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Step 3.3.1.3.1

Factor $2$ out of $4$ .

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

Step 3.3.1.3.2

Cancel the common factor.

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

Step 3.3.1.3.3

Rewrite the expression.

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

Step 3.3.1.4

Apply the distributive property.

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

Step 3.3.1.5

Cancel the common factor of $2$ .

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Step 3.3.1.5.1

Cancel the common factor.

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

Step 3.3.1.5.2

Rewrite the expression.

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

Step 3.3.1.6

Multiply $2$ by $2$ .

$3 y + 4 = 2 x$

Step 3.4

Subtract $4$ from both sides of the equation.

$3 y = 2 x - 4$

Step 3.5

Divide each term in $3 y = 2 x - 4$ by $3$ and simplify.

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Step 3.5.1

Divide each term in $3 y = 2 x - 4$ by $3$ .

$\frac{3 y}{3} = \frac{2 x}{3} + \frac{- 4}{3}$

Step 3.5.2

Simplify the left side.

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Step 3.5.2.1

Cancel the common factor of $3$ .

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Step 3.5.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{2 x}{3} + \frac{- 4}{3}$

Step 3.5.2.1.2

Divide $y$ by $1$ .

$y = \frac{2 x}{3} + \frac{- 4}{3}$

Step 3.5.3

Simplify the right side.

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Step 3.5.3.1

Move the negative in front of the fraction.

$y = \frac{2 x}{3} - \frac{4}{3}$

Step 4

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

$f^{- 1} (x) = \frac{2 x}{3} - \frac{4}{3}$

Step 5

Verify if $f^{- 1} (x) = \frac{2 x}{3} - \frac{4}{3}$ is the inverse of $f (x) = \frac{1}{2} \cdot (3 x + 4)$ .

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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 $f^{- 1} (\frac{1}{2} \cdot (3 x + 4))$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{1}{2} \cdot (3 x + 4)) = \frac{2 (\frac{1}{2} \cdot (3 x + 4))}{3} - \frac{4}{3}$

Step 5.2.3

Combine the numerators over the common denominator.

$f^{- 1} (\frac{1}{2} \cdot (3 x + 4)) = \frac{2 (\frac{1}{2} \cdot (3 x + 4)) - 4}{3}$

Step 5.2.4

Simplify each term.

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Step 5.2.4.1

Apply the distributive property.

$f^{- 1} (\frac{1}{2} \cdot (3 x + 4)) = \frac{2 (\frac{1}{2} \cdot (3 x) + \frac{1}{2} \cdot 4) - 4}{3}$

Step 5.2.4.2

Multiply $\frac{1}{2} (3 x)$ .

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Step 5.2.4.2.1

Combine $3$ and $\frac{1}{2}$ .

$f^{- 1} (\frac{1}{2} \cdot (3 x + 4)) = \frac{2 (\frac{3}{2} x + \frac{1}{2} \cdot 4) - 4}{3}$

Step 5.2.4.2.2

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

$f^{- 1} (\frac{1}{2} \cdot (3 x + 4)) = \frac{2 (\frac{3 x}{2} + \frac{1}{2} \cdot 4) - 4}{3}$

Step 5.2.4.3

Cancel the common factor of $2$ .

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Step 5.2.4.3.1

Factor $2$ out of $4$ .

$f^{- 1} (\frac{1}{2} \cdot (3 x + 4)) = \frac{2 (\frac{3 x}{2} + \frac{1}{2} \cdot (2 (2))) - 4}{3}$

Step 5.2.4.3.2

Cancel the common factor.

$f^{- 1} (\frac{1}{2} \cdot (3 x + 4)) = \frac{2 (\frac{3 x}{2} + \frac{1}{2} \cdot (2 \cdot 2)) - 4}{3}$

Step 5.2.4.3.3

Rewrite the expression.

$f^{- 1} (\frac{1}{2} \cdot (3 x + 4)) = \frac{2 (\frac{3 x}{2} + 2) - 4}{3}$

Step 5.2.4.4

Apply the distributive property.

$f^{- 1} (\frac{1}{2} \cdot (3 x + 4)) = \frac{2 (\frac{3 x}{2}) + 2 \cdot 2 - 4}{3}$

Step 5.2.4.5

Cancel the common factor of $2$ .

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Step 5.2.4.5.1

Cancel the common factor.

$f^{- 1} (\frac{1}{2} \cdot (3 x + 4)) = \frac{2 (\frac{3 x}{2}) + 2 \cdot 2 - 4}{3}$

Step 5.2.4.5.2

Rewrite the expression.

$f^{- 1} (\frac{1}{2} \cdot (3 x + 4)) = \frac{3 x + 2 \cdot 2 - 4}{3}$

Step 5.2.4.6

Multiply $2$ by $2$ .

$f^{- 1} (\frac{1}{2} \cdot (3 x + 4)) = \frac{3 x + 4 - 4}{3}$

Step 5.2.5

Simplify terms.

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Step 5.2.5.1

Combine the opposite terms in $3 x + 4 - 4$ .

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Step 5.2.5.1.1

Subtract $4$ from $4$ .

$f^{- 1} (\frac{1}{2} \cdot (3 x + 4)) = \frac{3 x + 0}{3}$

Step 5.2.5.1.2

Add $3 x$ and $0$ .

$f^{- 1} (\frac{1}{2} \cdot (3 x + 4)) = \frac{3 x}{3}$

Step 5.2.5.2

Cancel the common factor of $3$ .

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Step 5.2.5.2.1

Cancel the common factor.

$f^{- 1} (\frac{1}{2} \cdot (3 x + 4)) = \frac{3 x}{3}$

Step 5.2.5.2.2

Divide $x$ by $1$ .

$f^{- 1} (\frac{1}{2} \cdot (3 x + 4)) = x$

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 $f (\frac{2 x}{3} - \frac{4}{3})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{2 x}{3} - \frac{4}{3}) = \frac{1}{2} \cdot (3 (\frac{2 x}{3} - \frac{4}{3}) + 4)$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Apply the distributive property.

$f (\frac{2 x}{3} - \frac{4}{3}) = \frac{1}{2} \cdot (3 (\frac{2 x}{3}) + 3 (- \frac{4}{3}) + 4)$

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.

$f (\frac{2 x}{3} - \frac{4}{3}) = \frac{1}{2} \cdot (3 (\frac{2 x}{3}) + 3 (- \frac{4}{3}) + 4)$

Step 5.3.3.2.2

Rewrite the expression.

$f (\frac{2 x}{3} - \frac{4}{3}) = \frac{1}{2} \cdot (2 x + 3 (- \frac{4}{3}) + 4)$

Step 5.3.3.3

Cancel the common factor of $3$ .

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Step 5.3.3.3.1

Move the leading negative in $- \frac{4}{3}$ into the numerator.

$f (\frac{2 x}{3} - \frac{4}{3}) = \frac{1}{2} \cdot (2 x + 3 (\frac{- 4}{3}) + 4)$

Step 5.3.3.3.2

Cancel the common factor.

$f (\frac{2 x}{3} - \frac{4}{3}) = \frac{1}{2} \cdot (2 x + 3 (\frac{- 4}{3}) + 4)$

Step 5.3.3.3.3

Rewrite the expression.

$f (\frac{2 x}{3} - \frac{4}{3}) = \frac{1}{2} \cdot (2 x - 4 + 4)$

Step 5.3.4

Simplify terms.

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Step 5.3.4.1

Combine the opposite terms in $2 x - 4 + 4$ .

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Step 5.3.4.1.1

Add $- 4$ and $4$ .

$f (\frac{2 x}{3} - \frac{4}{3}) = \frac{1}{2} \cdot (2 x + 0)$

Step 5.3.4.1.2

Add $2 x$ and $0$ .

$f (\frac{2 x}{3} - \frac{4}{3}) = \frac{1}{2} \cdot (2 x)$

Step 5.3.4.2

Cancel the common factor of $2$ .

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Step 5.3.4.2.1

Factor $2$ out of $2 x$ .

$f (\frac{2 x}{3} - \frac{4}{3}) = \frac{1}{2} \cdot (2 (x))$

Step 5.3.4.2.2

Cancel the common factor.

$f (\frac{2 x}{3} - \frac{4}{3}) = \frac{1}{2} \cdot (2 x)$

Step 5.3.4.2.3

Rewrite the expression.

$f (\frac{2 x}{3} - \frac{4}{3}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{2 x}{3} - \frac{4}{3}$ is the inverse of $f (x) = \frac{1}{2} \cdot (3 x + 4)$ .

$f^{- 1} (x) = \frac{2 x}{3} - \frac{4}{3}$