Precalculus Examples

$f (x) = 10 x + 20$

Step 1

Write $f (x) = 10 x + 20$ as an equation.

$y = 10 x + 20$

Step 2

Interchange the variables.

$x = 10 y + 20$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $10 y + 20 = x$ .

$10 y + 20 = x$

Step 3.2

Subtract $20$ from both sides of the equation.

$10 y = x - 20$

Step 3.3

Divide each term in $10 y = x - 20$ by $10$ and simplify.

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Step 3.3.1

Divide each term in $10 y = x - 20$ by $10$ .

$\frac{10 y}{10} = \frac{x}{10} + \frac{- 20}{10}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $10$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{10 y}{10} = \frac{x}{10} + \frac{- 20}{10}$

Step 3.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{x}{10} + \frac{- 20}{10}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Divide $- 20$ by $10$ .

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

Step 4

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

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

Step 5

Verify if $f^{- 1} (x) = \frac{x}{10} - 2$ is the inverse of $f (x) = 10 x + 20$ .

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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 $f^{- 1} (10 x + 20)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (10 x + 20) = \frac{10 x + 20}{10} - 2$

Step 5.2.3

Cancel the common factor of $10 x + 20$ and $10$ .

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Step 5.2.3.1

Factor $10$ out of $10 x$ .

$f^{- 1} (10 x + 20) = \frac{10 (x) + 20}{10} - 2$

Step 5.2.3.2

Factor $10$ out of $20$ .

$f^{- 1} (10 x + 20) = \frac{10 (x) + 10 \cdot 2}{10} - 2$

Step 5.2.3.3

Factor $10$ out of $10 (x) + 10 (2)$ .

$f^{- 1} (10 x + 20) = \frac{10 (x + 2)}{10} - 2$

Step 5.2.3.4

Cancel the common factors.

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Step 5.2.3.4.1

Factor $10$ out of $10$ .

$f^{- 1} (10 x + 20) = \frac{10 (x + 2)}{10 (1)} - 2$

Step 5.2.3.4.2

Cancel the common factor.

$f^{- 1} (10 x + 20) = \frac{10 (x + 2)}{10 \cdot 1} - 2$

Step 5.2.3.4.3

Rewrite the expression.

$f^{- 1} (10 x + 20) = \frac{x + 2}{1} - 2$

Step 5.2.3.4.4

Divide $x + 2$ by $1$ .

$f^{- 1} (10 x + 20) = x + 2 - 2$

Step 5.2.4

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

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Step 5.2.4.1

Subtract $2$ from $2$ .

$f^{- 1} (10 x + 20) = x + 0$

Step 5.2.4.2

Add $x$ and $0$ .

$f^{- 1} (10 x + 20) = x$

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

$f (\frac{x}{10} - 2) = 10 (\frac{x}{10} - 2) + 20$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Apply the distributive property.

$f (\frac{x}{10} - 2) = 10 (\frac{x}{10}) + 10 \cdot - 2 + 20$

Step 5.3.3.2

Cancel the common factor of $10$ .

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Step 5.3.3.2.1

Cancel the common factor.

$f (\frac{x}{10} - 2) = 10 (\frac{x}{10}) + 10 \cdot - 2 + 20$

Step 5.3.3.2.2

Rewrite the expression.

$f (\frac{x}{10} - 2) = x + 10 \cdot - 2 + 20$

Step 5.3.3.3

Multiply $10$ by $- 2$ .

$f (\frac{x}{10} - 2) = x - 20 + 20$

Step 5.3.4

Combine the opposite terms in $x - 20 + 20$ .

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Step 5.3.4.1

Add $- 20$ and $20$ .

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

Step 5.3.4.2

Add $x$ and $0$ .

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

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x}{10} - 2$ is the inverse of $f (x) = 10 x + 20$ .

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