Finite Math Examples

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

Step 1

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

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{1}{2} y - b = x$ .

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

Step 3.2

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

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

Step 3.3

Add $b$ to both sides of the equation.

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

Step 3.4

Multiply both sides of the equation by $2$ .

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

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 $2$ .

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Step 3.5.1.1.1

Cancel the common factor.

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

Step 3.5.1.1.2

Rewrite the expression.

$y = 2 (x + b)$

Step 3.5.2

Simplify the right side.

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Step 3.5.2.1

Apply the distributive property.

$y = 2 x + 2 b$

Step 3.6

Reorder $2 x$ and $2 b$ .

$y = 2 b + 2 x$

Step 4

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

$f^{- 1} (x) = 2 b + 2 x$

Step 5

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

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

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

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

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

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

Step 5.2.3.2

Apply the distributive property.

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

Step 5.2.3.3

Cancel the common factor of $2$ .

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Step 5.2.3.3.1

Cancel the common factor.

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

Step 5.2.3.3.2

Rewrite the expression.

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

Step 5.2.3.4

Multiply $- 1$ by $2$ .

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

Step 5.2.4

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

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Step 5.2.4.1

Subtract $2 b$ from $2 b$ .

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

Step 5.2.4.2

Add $x$ and $0$ .

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

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

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

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Apply the distributive property.

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

Step 5.3.3.2

Cancel the common factor of $2$ .

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Step 5.3.3.2.1

Factor $2$ out of $2 b$ .

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

Step 5.3.3.2.2

Cancel the common factor.

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

Step 5.3.3.2.3

Rewrite the expression.

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

Step 5.3.3.3

Cancel the common factor of $2$ .

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Step 5.3.3.3.1

Factor $2$ out of $2 x$ .

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

Step 5.3.3.3.2

Cancel the common factor.

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

Step 5.3.3.3.3

Rewrite the expression.

$f (2 b + 2 x) = b + x - b$

Step 5.3.4

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

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Step 5.3.4.1

Subtract $b$ from $b$ .

$f (2 b + 2 x) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

$f (2 b + 2 x) = x$

Step 5.4

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

$f^{- 1} (x) = 2 b + 2 x$