Precalculus Examples

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

Step 1

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

$y = 3 x + 6$

Step 2

Interchange the variables.

$x = 3 y + 6$

Step 3

Solve for $y$ .

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Step 3.1

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

$3 y + 6 = x$

Step 3.2

Subtract $6$ from both sides of the equation.

$3 y = x - 6$

Step 3.3

Divide each term in $3 y = x - 6$ by $3$ and simplify.

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Step 3.3.1

Divide each term in $3 y = x - 6$ by $3$ .

$\frac{3 y}{3} = \frac{x}{3} + \frac{- 6}{3}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $3$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{x}{3} + \frac{- 6}{3}$

Step 3.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{x}{3} + \frac{- 6}{3}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Divide $- 6$ by $3$ .

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

Step 4

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

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

Step 5

Verify if $f^{- 1} (x) = \frac{x}{3} - 2$ is the inverse of $f (x) = 3 x + 6$ .

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

$f^{- 1} (3 x + 6) = \frac{3 x + 6}{3} - 2$

Step 5.2.3

Cancel the common factor of $3 x + 6$ and $3$ .

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Step 5.2.3.1

Factor $3$ out of $3 x$ .

$f^{- 1} (3 x + 6) = \frac{3 (x) + 6}{3} - 2$

Step 5.2.3.2

Factor $3$ out of $6$ .

$f^{- 1} (3 x + 6) = \frac{3 (x) + 3 \cdot 2}{3} - 2$

Step 5.2.3.3

Factor $3$ out of $3 (x) + 3 (2)$ .

$f^{- 1} (3 x + 6) = \frac{3 (x + 2)}{3} - 2$

Step 5.2.3.4

Cancel the common factors.

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Step 5.2.3.4.1

Factor $3$ out of $3$ .

$f^{- 1} (3 x + 6) = \frac{3 (x + 2)}{3 (1)} - 2$

Step 5.2.3.4.2

Cancel the common factor.

$f^{- 1} (3 x + 6) = \frac{3 (x + 2)}{3 \cdot 1} - 2$

Step 5.2.3.4.3

Rewrite the expression.

$f^{- 1} (3 x + 6) = \frac{x + 2}{1} - 2$

Step 5.2.3.4.4

Divide $x + 2$ by $1$ .

$f^{- 1} (3 x + 6) = x + 2 - 2$

Step 5.2.4

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

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Step 5.2.4.1

Subtract $2$ from $2$ .

$f^{- 1} (3 x + 6) = x + 0$

Step 5.2.4.2

Add $x$ and $0$ .

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

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

$f (\frac{x}{3} - 2) = 3 (\frac{x}{3} - 2) + 6$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Apply the distributive property.

$f (\frac{x}{3} - 2) = 3 (\frac{x}{3}) + 3 \cdot - 2 + 6$

Step 5.3.3.2

Cancel the common factor of $3$ .

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Step 5.3.3.2.1

Cancel the common factor.

$f (\frac{x}{3} - 2) = 3 (\frac{x}{3}) + 3 \cdot - 2 + 6$

Step 5.3.3.2.2

Rewrite the expression.

$f (\frac{x}{3} - 2) = x + 3 \cdot - 2 + 6$

Step 5.3.3.3

Multiply $3$ by $- 2$ .

$f (\frac{x}{3} - 2) = x - 6 + 6$

Step 5.3.4

Combine the opposite terms in $x - 6 + 6$ .

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Step 5.3.4.1

Add $- 6$ and $6$ .

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

Step 5.3.4.2

Add $x$ and $0$ .

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

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x}{3} - 2$ is the inverse of $f (x) = 3 x + 6$ .

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