Algebra Examples

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

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

Step 1

Write

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

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

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

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

Step 2

Interchange the variables.

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

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

Step 3

Solve for

y

$y$ .

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Step 3.1

Rewrite the equation as

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

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

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

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

Step 3.2

Combine

\frac{1}{3}

$\frac{1}{3}$ and

y

$y$ .

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

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

Step 3.3

Add

2

$2$ to both sides of the equation.

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

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

Step 3.4

Multiply both sides of the equation by

3

$3$ .

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

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

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

Cancel the common factor of

3

$3$ .

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Step 3.5.1.1.1

Cancel the common factor.

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

Step 3.5.1.1.2

Rewrite the expression.

$y = 3 (x + 2)$

Step 3.5.2

Simplify the right side.

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Step 3.5.2.1

Simplify

$3 (x + 2)$ .

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Step 3.5.2.1.1

Apply the distributive property.

$y = 3 x + 3 \cdot 2$

Step 3.5.2.1.2

Multiply

$3$ by

$2$ .

$y = 3 x + 6$

Step 4

Replace

$y$ with

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

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

Step 5

Verify if

$f^{- 1} (x) = 3 x + 6$ is the inverse of

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

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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

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

$f$ into

$f^{- 1}$ .

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

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Combine

$\frac{1}{3}$ and

$x$ .

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

Step 5.2.3.2

Apply the distributive property.

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

Step 5.2.3.3

Cancel the common factor of

$3$ .

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Step 5.2.3.3.1

Cancel the common factor.

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

Step 5.2.3.3.2

Rewrite the expression.

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

Step 5.2.3.4

Multiply

$3$ by

$- 2$ .

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

Step 5.2.4

Combine the opposite terms in

$x - 6 + 6$ .

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Step 5.2.4.1

Add

$- 6$ and

$6$ .

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

Step 5.2.4.2

Add

$x$ and

$0$ .

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

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

$f (3 x + 6)$ by substituting in the value of

$f^{- 1}$ into

$f$ .

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

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Apply the distributive property.

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

Step 5.3.3.2

Cancel the common factor of

$3$ .

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Step 5.3.3.2.1

Factor

$3$ out of

$3 x$ .

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

Step 5.3.3.2.2

Cancel the common factor.

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

Step 5.3.3.2.3

Rewrite the expression.

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

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

$6$ .

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

Step 5.3.3.3.2

Cancel the common factor.

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

Step 5.3.3.3.3

Rewrite the expression.

$f (3 x + 6) = x + 2 - 2$

Step 5.3.4

Combine the opposite terms in

$x + 2 - 2$ .

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Step 5.3.4.1

Subtract

$2$ from

$2$ .

$f (3 x + 6) = x + 0$

Step 5.3.4.2

Add

$x$ and

$0$ .

$f (3 x + 6) = x$

Step 5.4

Since

$f^{- 1} (f (x)) = x$ and

$f (f^{- 1} (x)) = x$ , then

$f^{- 1} (x) = 3 x + 6$ is the inverse of

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

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