Algebra Examples

$- 2 + \ln (x)$

Step 1

Interchange the variables.

$x = - 2 + \ln (y)$

Step 2

Solve for $y$ .

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Rewrite the equation as $- 2 + \ln (y) = x$ .

$- 2 + \ln (y) = x$

Add $2$ to both sides of the equation.

$\ln (y) = x + 2$

To solve for $y$ , rewrite the equation using properties of logarithms.

$e^{\ln (y)} = e^{x + 2}$

Rewrite $\ln (y) = x + 2$ 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 $b^{y} = x$ .

$e^{x + 2} = y$

Rewrite the equation as $y = e^{x + 2}$ .

$y = e^{x + 2}$

Step 3

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

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

Step 4

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

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

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

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Set up the composite result function.

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

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

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

Combine the opposite terms in $(- 2 + \ln (x)) + 2$ .

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Add $- 2$ and $2$ .

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

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

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

Exponentiation and log are inverse functions.

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

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

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Set up the composite result function.

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

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

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

Simplify each term.

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Use logarithm rules to move $x + 2$ out of the exponent.

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

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

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

Multiply $x + 2$ by $1$ .

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

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

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Add $- 2$ and $2$ .

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

Add $x$ and $0$ .

$f (e^{x + 2}) = x$

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

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