Finite Math Examples

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

Step 1

Write $f (x) = 3 x + 6 x$ as an equation.

$y = 3 x + 6 x$

Step 2

Interchange the variables.

$x = 3 y + 6 y$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $3 y + 6 y = x$ .

$3 y + 6 y = x$

Step 3.2

Add $3 y$ and $6 y$ .

$9 y = x$

Step 3.3

Divide each term in $9 y = x$ by $9$ and simplify.

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Step 3.3.1

Divide each term in $9 y = x$ by $9$ .

$\frac{9 y}{9} = \frac{x}{9}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $9$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{9 y}{9} = \frac{x}{9}$

Step 3.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{x}{9}$

Step 4

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

$f^{- 1} (x) = \frac{x}{9}$

Step 5

Verify if $f^{- 1} (x) = \frac{x}{9}$ is the inverse of $f (x) = 3 x + 6 x$ .

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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 $f^{- 1} (3 x + 6 x)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (3 x + 6 x) = \frac{3 x + 6 x}{9}$

Step 5.2.3

Add $3 x$ and $6 x$ .

$f^{- 1} (3 x + 6 x) = \frac{9 x}{9}$

Step 5.2.4

Cancel the common factor of $9$ .

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Step 5.2.4.1

Cancel the common factor.

$f^{- 1} (3 x + 6 x) = \frac{9 x}{9}$

Step 5.2.4.2

Divide $x$ by $1$ .

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

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 $f (\frac{x}{9})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x}{9}) = 3 (\frac{x}{9}) + 6 (\frac{x}{9})$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Cancel the common factor of $3$ .

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Step 5.3.3.1.1

Factor $3$ out of $9$ .

$f (\frac{x}{9}) = 3 (\frac{x}{3 (3)}) + 6 (\frac{x}{9})$

Step 5.3.3.1.2

Cancel the common factor.

$f (\frac{x}{9}) = 3 (\frac{x}{3 \cdot 3}) + 6 (\frac{x}{9})$

Step 5.3.3.1.3

Rewrite the expression.

$f (\frac{x}{9}) = \frac{x}{3} + 6 (\frac{x}{9})$

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 $6$ .

$f (\frac{x}{9}) = \frac{x}{3} + 3 (2) (\frac{x}{9})$

Step 5.3.3.2.2

Factor $3$ out of $9$ .

$f (\frac{x}{9}) = \frac{x}{3} + 3 \cdot (2 (\frac{x}{3 \cdot 3}))$

Step 5.3.3.2.3

Cancel the common factor.

$f (\frac{x}{9}) = \frac{x}{3} + 3 \cdot (2 (\frac{x}{3 \cdot 3}))$

Step 5.3.3.2.4

Rewrite the expression.

$f (\frac{x}{9}) = \frac{x}{3} + 2 (\frac{x}{3})$

Step 5.3.3.3

Combine $2$ and $\frac{x}{3}$ .

$f (\frac{x}{9}) = \frac{x}{3} + \frac{2 x}{3}$

Step 5.3.4

Simplify terms.

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Step 5.3.4.1

Combine the numerators over the common denominator.

$f (\frac{x}{9}) = \frac{x + 2 x}{3}$

Step 5.3.4.2

Add $x$ and $2 x$ .

$f (\frac{x}{9}) = \frac{3 x}{3}$

Step 5.3.4.3

Cancel the common factor of $3$ .

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Step 5.3.4.3.1

Cancel the common factor.

$f (\frac{x}{9}) = \frac{3 x}{3}$

Step 5.3.4.3.2

Divide $x$ by $1$ .

$f (\frac{x}{9}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x}{9}$ is the inverse of $f (x) = 3 x + 6 x$ .

$f^{- 1} (x) = \frac{x}{9}$