Precalculus Examples

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

Step 1

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

$y = 6 (x - 3)$

Step 2

Interchange the variables.

$x = 6 (y - 3)$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $6 (y - 3) = x$ .

$6 (y - 3) = x$

Step 3.2

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

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Step 3.2.1

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

$\frac{6 (y - 3)}{6} = \frac{x}{6}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $6$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{6 (y - 3)}{6} = \frac{x}{6}$

Step 3.2.2.1.2

Divide $y - 3$ by $1$ .

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

Step 3.3

Add $3$ to both sides of the equation.

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

Step 4

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

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

Step 5

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

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

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

Step 5.2.3

Cancel the common factor of $6$ .

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Step 5.2.3.1

Cancel the common factor.

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

Step 5.2.3.2

Divide $x - 3$ by $1$ .

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

Step 5.2.4

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

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Step 5.2.4.1

Add $- 3$ and $3$ .

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

Step 5.2.4.2

Add $x$ and $0$ .

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

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

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

Step 5.3.3

Combine the opposite terms in $(\frac{x}{6} + 3) - 3$ .

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Step 5.3.3.1

Subtract $3$ from $3$ .

$f (\frac{x}{6} + 3) = 6 (\frac{x}{6} + 0)$

Step 5.3.3.2

Add $\frac{x}{6}$ and $0$ .

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

Step 5.3.4

Cancel the common factor of $6$ .

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Step 5.3.4.1

Cancel the common factor.

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

Step 5.3.4.2

Rewrite the expression.

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

Step 5.4

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

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