Finite Math Examples

$f (x) = 5 (2 x + 3)$

Step 1

Write $f (x) = 5 (2 x + 3)$ as an equation.

$y = 5 (2 x + 3)$

Step 2

Interchange the variables.

$x = 5 (2 y + 3)$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $5 (2 y + 3) = x$ .

$5 (2 y + 3) = x$

Step 3.2

Divide each term in $5 (2 y + 3) = x$ by $5$ and simplify.

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Step 3.2.1

Divide each term in $5 (2 y + 3) = x$ by $5$ .

$\frac{5 (2 y + 3)}{5} = \frac{x}{5}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $5$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{5 (2 y + 3)}{5} = \frac{x}{5}$

Step 3.2.2.1.2

Divide $2 y + 3$ by $1$ .

$2 y + 3 = \frac{x}{5}$

Step 3.3

Subtract $3$ from both sides of the equation.

$2 y = \frac{x}{5} - 3$

Step 3.4

Divide each term in $2 y = \frac{x}{5} - 3$ by $2$ and simplify.

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Step 3.4.1

Divide each term in $2 y = \frac{x}{5} - 3$ by $2$ .

$\frac{2 y}{2} = \frac{\frac{x}{5}}{2} + \frac{- 3}{2}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $2$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{\frac{x}{5}}{2} + \frac{- 3}{2}$

Step 3.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{\frac{x}{5}}{2} + \frac{- 3}{2}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Simplify each term.

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Step 3.4.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{x}{5} \cdot \frac{1}{2} + \frac{- 3}{2}$

Step 3.4.3.1.2

Multiply $\frac{x}{5} \cdot \frac{1}{2}$ .

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Step 3.4.3.1.2.1

Multiply $\frac{x}{5}$ by $\frac{1}{2}$ .

$y = \frac{x}{5 \cdot 2} + \frac{- 3}{2}$

Step 3.4.3.1.2.2

Multiply $5$ by $2$ .

$y = \frac{x}{10} + \frac{- 3}{2}$

Step 3.4.3.1.3

Move the negative in front of the fraction.

$y = \frac{x}{10} - \frac{3}{2}$

Step 4

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

$f^{- 1} (x) = \frac{x}{10} - \frac{3}{2}$

Step 5

Verify if $f^{- 1} (x) = \frac{x}{10} - \frac{3}{2}$ is the inverse of $f (x) = 5 (2 x + 3)$ .

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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 $f^{- 1} (5 (2 x + 3))$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (5 (2 x + 3)) = \frac{5 (2 x + 3)}{10} - \frac{3}{2}$

Step 5.2.3

Cancel the common factors.

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Step 5.2.3.1

Factor $5$ out of $10$ .

$f^{- 1} (5 (2 x + 3)) = \frac{5 (2 x + 3)}{5 \cdot 2} - \frac{3}{2}$

Step 5.2.3.2

Cancel the common factor.

$f^{- 1} (5 (2 x + 3)) = \frac{5 (2 x + 3)}{5 \cdot 2} - \frac{3}{2}$

Step 5.2.3.3

Rewrite the expression.

$f^{- 1} (5 (2 x + 3)) = \frac{2 x + 3}{2} - \frac{3}{2}$

Step 5.2.4

Combine the numerators over the common denominator.

$f^{- 1} (5 (2 x + 3)) = \frac{2 x + 3 - 3}{2}$

Step 5.2.5

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

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Step 5.2.5.1

Subtract $3$ from $3$ .

$f^{- 1} (5 (2 x + 3)) = \frac{2 x + 0}{2}$

Step 5.2.5.2

Add $2 x$ and $0$ .

$f^{- 1} (5 (2 x + 3)) = \frac{2 x}{2}$

Step 5.2.6

Cancel the common factor of $2$ .

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Step 5.2.6.1

Cancel the common factor.

$f^{- 1} (5 (2 x + 3)) = \frac{2 x}{2}$

Step 5.2.6.2

Divide $x$ by $1$ .

$f^{- 1} (5 (2 x + 3)) = x$

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 $f (\frac{x}{10} - \frac{3}{2})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x}{10} - \frac{3}{2}) = 5 (2 (\frac{x}{10} - \frac{3}{2}) + 3)$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Apply the distributive property.

$f (\frac{x}{10} - \frac{3}{2}) = 5 (2 (\frac{x}{10}) + 2 (- \frac{3}{2}) + 3)$

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 $10$ .

$f (\frac{x}{10} - \frac{3}{2}) = 5 (2 (\frac{x}{2 (5)}) + 2 (- \frac{3}{2}) + 3)$

Step 5.3.3.2.2

Cancel the common factor.

$f (\frac{x}{10} - \frac{3}{2}) = 5 (2 (\frac{x}{2 \cdot 5}) + 2 (- \frac{3}{2}) + 3)$

Step 5.3.3.2.3

Rewrite the expression.

$f (\frac{x}{10} - \frac{3}{2}) = 5 (\frac{x}{5} + 2 (- \frac{3}{2}) + 3)$

Step 5.3.3.3

Cancel the common factor of $2$ .

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Step 5.3.3.3.1

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

$f (\frac{x}{10} - \frac{3}{2}) = 5 (\frac{x}{5} + 2 (\frac{- 3}{2}) + 3)$

Step 5.3.3.3.2

Cancel the common factor.

$f (\frac{x}{10} - \frac{3}{2}) = 5 (\frac{x}{5} + 2 (\frac{- 3}{2}) + 3)$

Step 5.3.3.3.3

Rewrite the expression.

$f (\frac{x}{10} - \frac{3}{2}) = 5 (\frac{x}{5} - 3 + 3)$

Step 5.3.4

Simplify terms.

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Step 5.3.4.1

Combine the opposite terms in $\frac{x}{5} - 3 + 3$ .

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Step 5.3.4.1.1

Add $- 3$ and $3$ .

$f (\frac{x}{10} - \frac{3}{2}) = 5 (\frac{x}{5} + 0)$

Step 5.3.4.1.2

Add $\frac{x}{5}$ and $0$ .

$f (\frac{x}{10} - \frac{3}{2}) = 5 (\frac{x}{5})$

Step 5.3.4.2

Cancel the common factor of $5$ .

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Step 5.3.4.2.1

Cancel the common factor.

$f (\frac{x}{10} - \frac{3}{2}) = 5 (\frac{x}{5})$

Step 5.3.4.2.2

Rewrite the expression.

$f (\frac{x}{10} - \frac{3}{2}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x}{10} - \frac{3}{2}$ is the inverse of $f (x) = 5 (2 x + 3)$ .

$f^{- 1} (x) = \frac{x}{10} - \frac{3}{2}$