Finite Math Examples

$f (x) = m x + b$

Step 1

Write $f (x) = m x + b$ as an equation.

$y = m x + b$

Step 2

Interchange the variables.

$m = y x + b$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $y x + b = m$ .

$y x + b = m$

Step 3.2

Subtract $b$ from both sides of the equation.

$y x = m - b$

Step 3.3

Divide each term in $y x = m - b$ by $x$ and simplify.

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Step 3.3.1

Divide each term in $y x = m - b$ by $x$ .

$\frac{y x}{x} = \frac{m}{x} + \frac{- b}{x}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $x$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{y x}{x} = \frac{m}{x} + \frac{- b}{x}$

Step 3.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{m}{x} + \frac{- b}{x}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Move the negative in front of the fraction.

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

Step 4

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

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

Step 5

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

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Step 5.1

To verify the inverse, check if $f^{- 1} (f (m)) = m$ and $f (f^{- 1} (m)) = m$ .

Step 5.2

Evaluate $f^{- 1} (f (m))$ .

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Step 5.2.1

Set up the composite result function.

$f^{- 1} (f (m))$

Step 5.2.2

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

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

Step 5.2.3

Combine the numerators over the common denominator.

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

Step 5.2.4

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

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Step 5.2.4.1

Subtract $b$ from $b$ .

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

Step 5.2.4.2

Add $m x$ and $0$ .

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

Step 5.2.5

Cancel the common factor of $x$ .

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Step 5.2.5.1

Cancel the common factor.

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

Step 5.2.5.2

Divide $m$ by $1$ .

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

Step 5.3

Evaluate $f (f^{- 1} (m))$ .

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Step 5.3.1

Set up the composite result function.

$f (f^{- 1} (m))$

Step 5.3.2

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

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

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Apply the distributive property.

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

Step 5.3.3.2

Cancel the common factor of $x$ .

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Step 5.3.3.2.1

Cancel the common factor.

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

Step 5.3.3.2.2

Rewrite the expression.

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

Step 5.3.3.3

Cancel the common factor of $x$ .

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Step 5.3.3.3.1

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

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

Step 5.3.3.3.2

Cancel the common factor.

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

Step 5.3.3.3.3

Rewrite the expression.

$f (\frac{m}{x} - \frac{b}{x}) = m - b + b$

Step 5.3.4

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

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Step 5.3.4.1

Add $- b$ and $b$ .

$f (\frac{m}{x} - \frac{b}{x}) = m + 0$

Step 5.3.4.2

Add $m$ and $0$ .

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

Step 5.4

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

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