Find the Inverse e^(x-4)

$e^{x - 4}$

Step 1

Interchange the variables.

$x = e^{y - 4}$

Step 2

Solve for $y$ .

Rewrite the equation as $e^{y - 4} = x$ .

$e^{y - 4} = x$

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{y - 4}) = \ln (x)$

Expand the left side.

Expand $\ln (e^{y - 4})$ by moving $y - 4$ outside the logarithm.

$(y - 4) \ln (e) = \ln (x)$

The natural logarithm of $e$ is $1$ .

$(y - 4) \cdot 1 = \ln (x)$

Multiply $y - 4$ by $1$ .

$y - 4 = \ln (x)$

Add $4$ to both sides of the equation.

$y = \ln (x) + 4$

Step 3

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

$f^{- 1} (x) = \ln (x) + 4$

Step 4

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

To verify the inverse, check if $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ .

Evaluate $f^{- 1} (f (x))$ .

Set up the composite result function.

$f^{- 1} (f (x))$

Evaluate $f^{- 1} (e^{x - 4})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (e^{x - 4}) = \ln (e^{x - 4}) + 4$

Simplify each term.

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

$f^{- 1} (e^{x - 4}) = (x - 4) \ln (e) + 4$

The natural logarithm of $e$ is $1$ .

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

Multiply $x - 4$ by $1$ .

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

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

Add $- 4$ and $4$ .

$f^{- 1} (e^{x - 4}) = x + 0$

Add $x$ and $0$ .

$f^{- 1} (e^{x - 4}) = x$

Evaluate $f (f^{- 1} (x))$ .

Set up the composite result function.

$f (f^{- 1} (x))$

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

$f (\ln (x) + 4) = e^{(\ln (x) + 4) - 4}$

Combine the opposite terms in $(\ln (x) + 4) - 4$ .

Subtract $4$ from $4$ .

$f (\ln (x) + 4) = e^{\ln (x) + 0}$

Add $\ln (x)$ and $0$ .

$f (\ln (x) + 4) = e^{\ln (x)}$

Exponentiation and log are inverse functions.

$f (\ln (x) + 4) = x$

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

$f^{- 1} (x) = \ln (x) + 4$

Find the Inverse e^(x-4)

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