Algebra Examples

$f (x) = e^{x} - 3$

Step 1

Write $f (x) = e^{x} - 3$ as an equation.

$y = e^{x} - 3$

Step 2

Interchange the variables.

$x = e^{y} - 3$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $e^{y} - 3 = x$ .

$e^{y} - 3 = x$

Step 3.2

Add $3$ to both sides of the equation.

$e^{y} = x + 3$

Step 3.3

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

$\ln (e^{y}) = \ln (x + 3)$

Step 3.4

Expand the left side.

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Step 3.4.1

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

$y \ln (e) = \ln (x + 3)$

Step 3.4.2

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

$y \cdot 1 = \ln (x + 3)$

Step 3.4.3

Multiply $y$ by $1$ .

$y = \ln (x + 3)$

Step 4

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

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

Step 5

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

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

$f^{- 1} (e^{x} - 3) = \ln ((e^{x} - 3) + 3)$

Step 5.2.3

Combine the opposite terms in $(e^{x} - 3) + 3$ .

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Step 5.2.3.1

Add $- 3$ and $3$ .

$f^{- 1} (e^{x} - 3) = \ln (e^{x} + 0)$

Step 5.2.3.2

Add $e^{x}$ and $0$ .

$f^{- 1} (e^{x} - 3) = \ln (e^{x})$

Step 5.2.4

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

$f^{- 1} (e^{x} - 3) = x \ln (e)$

Step 5.2.5

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

$f^{- 1} (e^{x} - 3) = x \cdot 1$

Step 5.2.6

Multiply $x$ by $1$ .

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

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

$f (\ln (x + 3)) = e^{\ln (x + 3)} - 3$

Step 5.3.3

Exponentiation and log are inverse functions.

$f (\ln (x + 3)) = x + 3 - 3$

Step 5.3.4

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

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Step 5.3.4.1

Subtract $3$ from $3$ .

$f (\ln (x + 3)) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

$f (\ln (x + 3)) = x$

Step 5.4

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

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