Precalculus Examples

$f (x) = \log_{3} (x - 2) - 4$

Step 1

Write $f (x) = \log_{3} (x - 2) - 4$ as an equation.

$y = \log_{3} (x - 2) - 4$

Step 2

Interchange the variables.

$x = \log_{3} (y - 2) - 4$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\log_{3} (y - 2) - 4 = x$ .

$\log_{3} (y - 2) - 4 = x$

Step 3.2

Add $4$ to both sides of the equation.

$\log_{3} (y - 2) = x + 4$

Step 3.3

Rewrite $\log_{3} (y - 2) = x + 4$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$3^{x + 4} = y - 2$

Step 3.4

Solve for $y$ .

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Step 3.4.1

Rewrite the equation as $y - 2 = 3^{x + 4}$ .

$y - 2 = 3^{x + 4}$

Step 3.4.2

Add $2$ to both sides of the equation.

$y = 3^{x + 4} + 2$

Step 4

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

$f^{- 1} (x) = 3^{x + 4} + 2$

Step 5

Verify if $f^{- 1} (x) = 3^{x + 4} + 2$ is the inverse of $f (x) = \log_{3} (x - 2) - 4$ .

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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 $f^{- 1} (\log_{3} (x - 2) - 4)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\log_{3} (x - 2) - 4) = 3^{(\log_{3} (x - 2) - 4) + 4} + 2$

Step 5.2.3

Combine the opposite terms in $3^{(\log_{3} (x - 2) - 4) + 4} + 2$ .

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Step 5.2.3.1

Add $- 4$ and $4$ .

$f^{- 1} (\log_{3} (x - 2) - 4) = 3^{\log_{3} (x - 2) + 0} + 2$

Step 5.2.3.2

Add $\log_{3} (x - 2)$ and $0$ .

$f^{- 1} (\log_{3} (x - 2) - 4) = 3^{\log_{3} (x - 2)} + 2$

Step 5.2.4

Exponentiation and log are inverse functions.

$f^{- 1} (\log_{3} (x - 2) - 4) = x - 2 + 2$

Step 5.2.5

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

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Step 5.2.5.1

Add $- 2$ and $2$ .

$f^{- 1} (\log_{3} (x - 2) - 4) = x + 0$

Step 5.2.5.2

Add $x$ and $0$ .

$f^{- 1} (\log_{3} (x - 2) - 4) = x$

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 $f (3^{x + 4} + 2)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (3^{x + 4} + 2) = \log_{3} ((3^{x + 4} + 2) - 2) - 4$

Step 5.3.3

Combine the opposite terms in $\log_{3} ((3^{x + 4} + 2) - 2) - 4$ .

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Step 5.3.3.1

Subtract $2$ from $2$ .

$f (3^{x + 4} + 2) = \log_{3} (3^{x + 4} + 0) - 4$

Step 5.3.3.2

Add $3^{x + 4}$ and $0$ .

$f (3^{x + 4} + 2) = \log_{3} (3^{x + 4}) - 4$

Step 5.3.4

Simplify each term.

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Step 5.3.4.1

Use logarithm rules to move $x + 4$ out of the exponent.

$f (3^{x + 4} + 2) = (x + 4) \log_{3} (3) - 4$

Step 5.3.4.2

Logarithm base $3$ of $3$ is $1$ .

$f (3^{x + 4} + 2) = (x + 4) \cdot 1 - 4$

Step 5.3.4.3

Multiply $x + 4$ by $1$ .

$f (3^{x + 4} + 2) = x + 4 - 4$

Step 5.3.5

Combine the opposite terms in $x + 4 - 4$ .

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Step 5.3.5.1

Subtract $4$ from $4$ .

$f (3^{x + 4} + 2) = x + 0$

Step 5.3.5.2

Add $x$ and $0$ .

$f (3^{x + 4} + 2) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 3^{x + 4} + 2$ is the inverse of $f (x) = \log_{3} (x - 2) - 4$ .

$f^{- 1} (x) = 3^{x + 4} + 2$