Finite Math Examples

$f (x) = 15000 + 1.5 x$

Step 1

Write $f (x) = 15000 + 1.5 x$ as an equation.

$y = 15000 + 1.5 x$

Step 2

Interchange the variables.

$x = 15000 + 1.5 y$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $15000 + 1.5 y = x$ .

$15000 + 1.5 y = x$

Step 3.2

Subtract $15000$ from both sides of the equation.

$1.5 y = x - 15000$

Step 3.3

Divide each term in $1.5 y = x - 15000$ by $1.5$ and simplify.

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Step 3.3.1

Divide each term in $1.5 y = x - 15000$ by $1.5$ .

$\frac{1.5 y}{1.5} = \frac{x}{1.5} + \frac{- 15000}{1.5}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $1.5$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{1.5 y}{1.5} = \frac{x}{1.5} + \frac{- 15000}{1.5}$

Step 3.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{x}{1.5} + \frac{- 15000}{1.5}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Simplify each term.

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Step 3.3.3.1.1

Multiply by $1$ .

$y = \frac{1 x}{1.5} + \frac{- 15000}{1.5}$

Step 3.3.3.1.2

Factor $1.5$ out of $1.5$ .

$y = \frac{1 x}{1.5 (1)} + \frac{- 15000}{1.5}$

Step 3.3.3.1.3

Separate fractions.

$y = \frac{1}{1.5} \cdot \frac{x}{1} + \frac{- 15000}{1.5}$

Step 3.3.3.1.4

Divide $1$ by $1.5$ .

$y = 0.\overline{6} \frac{x}{1} + \frac{- 15000}{1.5}$

Step 3.3.3.1.5

Divide $x$ by $1$ .

$y = 0.\overline{6} x + \frac{- 15000}{1.5}$

Step 3.3.3.1.6

Divide $- 15000$ by $1.5$ .

$y = 0.\overline{6} x - 10000$

Step 4

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

$f^{- 1} (x) = 0.\overline{6} x - 10000$

Step 5

Verify if $f^{- 1} (x) = 0.\overline{6} x - 10000$ is the inverse of $f (x) = 15000 + 1.5 x$ .

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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 $f^{- 1} (15000 + 1.5 x)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (15000 + 1.5 x) = 0.\overline{6} (15000 + 1.5 x) - 10000$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Apply the distributive property.

$f^{- 1} (15000 + 1.5 x) = 0.\overline{6} \cdot 15000 + 0.\overline{6} (1.5 x) - 10000$

Step 5.2.3.2

Multiply $0.\overline{6}$ by $15000$ .

$f^{- 1} (15000 + 1.5 x) = 10000 + 0.\overline{6} (1.5 x) - 10000$

Step 5.2.3.3

Multiply $0.\overline{6} (1.5 x)$ .

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Step 5.2.3.3.1

Multiply $1.5$ by $0.\overline{6}$ .

$f^{- 1} (15000 + 1.5 x) = 10000 + 1 x - 10000$

Step 5.2.3.3.2

Multiply $x$ by $1$ .

$f^{- 1} (15000 + 1.5 x) = 10000 + x - 10000$

Step 5.2.4

Simplify by subtracting numbers.

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Step 5.2.4.1

Subtract $10000$ from $10000$ .

$f^{- 1} (15000 + 1.5 x) = x + 0$

Step 5.2.4.2

Add $x$ and $0$ .

$f^{- 1} (15000 + 1.5 x) = x$

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 $f (0.\overline{6} x - 10000)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (0.\overline{6} x - 10000) = 15000 + 1.5 (0.\overline{6} x - 10000)$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Apply the distributive property.

$f (0.\overline{6} x - 10000) = 15000 + 1.5 (0.\overline{6} x) + 1.5 \cdot - 10000$

Step 5.3.3.2

Multiply $1.5 (0.\overline{6} x)$ .

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Step 5.3.3.2.1

Multiply $0.\overline{6}$ by $1.5$ .

$f (0.\overline{6} x - 10000) = 15000 + 1 x + 1.5 \cdot - 10000$

Step 5.3.3.2.2

Multiply $x$ by $1$ .

$f (0.\overline{6} x - 10000) = 15000 + x + 1.5 \cdot - 10000$

Step 5.3.3.3

Multiply $1.5$ by $- 10000$ .

$f (0.\overline{6} x - 10000) = 15000 + x - 15000$

Step 5.3.4

Simplify by subtracting numbers.

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Step 5.3.4.1

Subtract $15000$ from $15000$ .

$f (0.\overline{6} x - 10000) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

$f (0.\overline{6} x - 10000) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 0.\overline{6} x - 10000$ is the inverse of $f (x) = 15000 + 1.5 x$ .

$f^{- 1} (x) = 0.\overline{6} x - 10000$