Algebra Examples

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

Step 1

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

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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Step 3.1

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

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

Step 3.2

Multiply both sides of the equation by $2$ .

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

Step 3.3

Simplify the left side.

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Step 3.3.1

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

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Step 3.3.1.1

Apply the distributive property.

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

Step 3.3.1.2

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

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

Step 3.3.1.3

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

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Step 3.3.1.3.1

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

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

Step 3.3.1.3.2

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

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

Step 3.3.1.4

Move the negative in front of the fraction.

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

Step 3.3.1.5

Apply the distributive property.

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

Step 3.3.1.6

Cancel the common factor of $2$ .

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Step 3.3.1.6.1

Cancel the common factor.

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

Step 3.3.1.6.2

Rewrite the expression.

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

Step 3.3.1.7

Cancel the common factor of $2$ .

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Step 3.3.1.7.1

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

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

Step 3.3.1.7.2

Cancel the common factor.

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

Step 3.3.1.7.3

Rewrite the expression.

$3 - 3 y = 2 x$

Step 3.4

Subtract $3$ from both sides of the equation.

$- 3 y = 2 x - 3$

Step 3.5

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

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Step 3.5.1

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

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

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.

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

Step 3.5.2.1.2

Divide $y$ by $1$ .

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

Step 3.5.3

Simplify the right side.

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Step 3.5.3.1

Simplify each term.

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Step 3.5.3.1.1

Move the negative in front of the fraction.

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

Step 3.5.3.1.2

Divide $- 3$ by $- 3$ .

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

Step 4

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

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

Step 5

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

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

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

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Cancel the common factor of $3 - 3 x$ and $3$ .

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Step 5.2.3.1.1

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

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

Step 5.2.3.1.2

Cancel the common factors.

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Step 5.2.3.1.2.1

Factor $3$ out of $3$ .

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

Step 5.2.3.1.2.2

Cancel the common factor.

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

Step 5.2.3.1.2.3

Rewrite the expression.

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

Step 5.2.3.1.2.4

Divide $2 (\frac{1}{2} \cdot (1 - x))$ by $1$ .

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

Step 5.2.3.2

Apply the distributive property.

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

Step 5.2.3.3

Multiply $\frac{1}{2}$ by $1$ .

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

Step 5.2.3.4

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

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

Step 5.2.3.5

Apply the distributive property.

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

Step 5.2.3.6

Cancel the common factor of $2$ .

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Step 5.2.3.6.1

Cancel the common factor.

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

Step 5.2.3.6.2

Rewrite the expression.

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

Step 5.2.3.7

Cancel the common factor of $2$ .

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Step 5.2.3.7.1

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

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

Step 5.2.3.7.2

Cancel the common factor.

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

Step 5.2.3.7.3

Rewrite the expression.

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

Step 5.2.3.8

Apply the distributive property.

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

Step 5.2.3.9

Multiply $- 1$ by $1$ .

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

Step 5.2.3.10

Multiply $- - x$ .

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Step 5.2.3.10.1

Multiply $- 1$ by $- 1$ .

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

Step 5.2.3.10.2

Multiply $x$ by $1$ .

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

Step 5.2.4

Combine the opposite terms in $- 1 + x + 1$ .

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Step 5.2.4.1

Add $- 1$ and $1$ .

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

Step 5.2.4.2

Add $x$ and $0$ .

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

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

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

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Apply the distributive property.

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

Step 5.3.3.2

Cancel the common factor of $3$ .

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Step 5.3.3.2.1

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

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

Step 5.3.3.2.2

Factor $3$ out of $- 3$ .

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

Step 5.3.3.2.3

Cancel the common factor.

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

Step 5.3.3.2.4

Rewrite the expression.

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

Step 5.3.3.3

Multiply $- 2$ by $- 1$ .

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

Step 5.3.3.4

Multiply $- 3$ by $1$ .

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

Step 5.3.4

Simplify terms.

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Step 5.3.4.1

Combine the opposite terms in $3 + 2 x - 3$ .

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Step 5.3.4.1.1

Subtract $3$ from $3$ .

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

Step 5.3.4.1.2

Add $2 x$ and $0$ .

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

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 $2 x$ .

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

Step 5.3.4.2.2

Cancel the common factor.

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

Step 5.3.4.2.3

Rewrite the expression.

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

Step 5.4

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

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