Find the Inverse f(x)=2+4/(e^(x-6))

$f (x) = 2 + \frac{4}{e^{x - 6}}$

Step 1

Write $f (x) = 2 + \frac{4}{e^{x - 6}}$ as an equation.

$y = 2 + \frac{4}{e^{x - 6}}$

Step 2

Interchange the variables.

$x = 2 + \frac{4}{e^{y - 6}}$

Step 3

Solve for $y$ .

Rewrite the equation as $2 + \frac{4}{e^{y - 6}} = x$ .

$2 + \frac{4}{e^{y - 6}} = x$

Subtract $2$ from both sides of the equation.

$\frac{4}{e^{y - 6}} = x - 2$

Multiply both sides by $e^{y - 6}$ .

$\frac{4}{e^{y - 6}} e^{y - 6} = (x - 2) e^{y - 6}$

Simplify.

Simplify the left side.

Cancel the common factor of $e^{y - 6}$ .

Cancel the common factor.

$\frac{4}{e^{y - 6}} e^{y - 6} = (x - 2) e^{y - 6}$

Rewrite the expression.

$4 = (x - 2) e^{y - 6}$

Simplify the right side.

Apply the distributive property.

$4 = x e^{y - 6} - 2 e^{y - 6}$

Solve for $y$ .

Rewrite the equation as $x e^{y - 6} - 2 e^{y - 6} = 4$ .

$x e^{y - 6} - 2 e^{y - 6} = 4$

Factor $e^{y - 6}$ out of $x e^{y - 6} - 2 e^{y - 6}$ .

Factor $e^{y - 6}$ out of $x e^{y - 6}$ .

$e^{y - 6} x - 2 e^{y - 6} = 4$

Factor $e^{y - 6}$ out of $- 2 e^{y - 6}$ .

$e^{y - 6} x + e^{y - 6} \cdot - 2 = 4$

Factor $e^{y - 6}$ out of $e^{y - 6} x + e^{y - 6} \cdot - 2$ .

$e^{y - 6} (x - 2) = 4$

Divide each term in $e^{y - 6} (x - 2) = 4$ by $x - 2$ and simplify.

Divide each term in $e^{y - 6} (x - 2) = 4$ by $x - 2$ .

$\frac{e^{y - 6} (x - 2)}{x - 2} = \frac{4}{x - 2}$

Simplify the left side.

Cancel the common factor of $x - 2$ .

Cancel the common factor.

$\frac{e^{y - 6} (x - 2)}{x - 2} = \frac{4}{x - 2}$

Divide $e^{y - 6}$ by $1$ .

$e^{y - 6} = \frac{4}{x - 2}$

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

$\ln (e^{y - 6}) = \ln (\frac{4}{x - 2})$

Expand the left side.

Expand $\ln (e^{y - 6})$ by moving $y - 6$ outside the logarithm.

$(y - 6) \ln (e) = \ln (\frac{4}{x - 2})$

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

$(y - 6) \cdot 1 = \ln (\frac{4}{x - 2})$

Multiply $y - 6$ by $1$ .

$y - 6 = \ln (\frac{4}{x - 2})$

Add $6$ to both sides of the equation.

$y = \ln (\frac{4}{x - 2}) + 6$

Step 4

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

$f^{- 1} (x) = \ln (\frac{4}{x - 2}) + 6$

Step 5

Verify if $f^{- 1} (x) = \ln (\frac{4}{x - 2}) + 6$ is the inverse of $f (x) = 2 + \frac{4}{e^{x - 6}}$ .

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 $f^{- 1} (2 + \frac{4}{e^{x - 6}})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (2 + \frac{4}{e^{x - 6}}) = \ln (\frac{4}{(2 + \frac{4}{e^{x - 6}}) - 2}) + 6$

Combine the opposite terms in $\ln (\frac{4}{(2 + \frac{4}{e^{x - 6}}) - 2}) + 6$ .

Subtract $2$ from $2$ .

$f^{- 1} (2 + \frac{4}{e^{x - 6}}) = \ln (\frac{4}{0 + \frac{4}{e^{x - 6}}}) + 6$

Add $0$ and $\frac{4}{e^{x - 6}}$ .

$f^{- 1} (2 + \frac{4}{e^{x - 6}}) = \ln (\frac{4}{\frac{4}{e^{x - 6}}}) + 6$

Simplify each term.

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (2 + \frac{4}{e^{x - 6}}) = \ln (4 (\frac{e^{x - 6}}{4})) + 6$

Cancel the common factor of $4$ .

Cancel the common factor.

$f^{- 1} (2 + \frac{4}{e^{x - 6}}) = \ln (4 (\frac{e^{x - 6}}{4})) + 6$

Rewrite the expression.

$f^{- 1} (2 + \frac{4}{e^{x - 6}}) = \ln (e^{x - 6}) + 6$

Use logarithm rules to move $x - 6$ out of the exponent.

$f^{- 1} (2 + \frac{4}{e^{x - 6}}) = (x - 6) \ln (e) + 6$

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

$f^{- 1} (2 + \frac{4}{e^{x - 6}}) = (x - 6) \cdot 1 + 6$

Multiply $x - 6$ by $1$ .

$f^{- 1} (2 + \frac{4}{e^{x - 6}}) = x - 6 + 6$

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

Add $- 6$ and $6$ .

$f^{- 1} (2 + \frac{4}{e^{x - 6}}) = x + 0$

Add $x$ and $0$ .

$f^{- 1} (2 + \frac{4}{e^{x - 6}}) = x$

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

Set up the composite result function.

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

Evaluate $f (\ln (\frac{4}{x - 2}) + 6)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\ln (\frac{4}{x - 2}) + 6) = 2 + \frac{4}{e^{(\ln (\frac{4}{x - 2}) + 6) - 6}}$

Combine the opposite terms in $2 + \frac{4}{e^{(\ln (\frac{4}{x - 2}) + 6) - 6}}$ .

Subtract $6$ from $6$ .

$f (\ln (\frac{4}{x - 2}) + 6) = 2 + \frac{4}{e^{\ln (\frac{4}{x - 2}) + 0}}$

Add $\ln (\frac{4}{x - 2})$ and $0$ .

$f (\ln (\frac{4}{x - 2}) + 6) = 2 + \frac{4}{e^{\ln (\frac{4}{x - 2})}}$

Simplify each term.

Exponentiation and log are inverse functions.

$f (\ln (\frac{4}{x - 2}) + 6) = 2 + \frac{4}{\frac{4}{x - 2}}$

Multiply the numerator by the reciprocal of the denominator.

$f (\ln (\frac{4}{x - 2}) + 6) = 2 + 4 (\frac{x - 2}{4})$

Cancel the common factor of $4$ .

Cancel the common factor.

$f (\ln (\frac{4}{x - 2}) + 6) = 2 + 4 (\frac{x - 2}{4})$

Rewrite the expression.

$f (\ln (\frac{4}{x - 2}) + 6) = 2 + x - 2$

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

Subtract $2$ from $2$ .

$f (\ln (\frac{4}{x - 2}) + 6) = x + 0$

Add $x$ and $0$ .

$f (\ln (\frac{4}{x - 2}) + 6) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \ln (\frac{4}{x - 2}) + 6$ is the inverse of $f (x) = 2 + \frac{4}{e^{x - 6}}$ .

$f^{- 1} (x) = \ln (\frac{4}{x - 2}) + 6$

Find the Inverse f(x)=2+4/(e^(x-6))

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