Calculus Examples

$f (x) = e^{- 8 x}$

Step 1

Write $f (x) = e^{- 8 x}$ as an equation.

$y = e^{- 8 x}$

Step 2

Interchange the variables.

$x = e^{- 8 y}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $e^{- 8 y} = x$ .

$e^{- 8 y} = x$

Step 3.2

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

$\ln (e^{- 8 y}) = \ln (x)$

Step 3.3

Expand the left side.

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Step 3.3.1

Expand $\ln (e^{- 8 y})$ by moving $- 8 y$ outside the logarithm.

$- 8 y \ln (e) = \ln (x)$

Step 3.3.2

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

$- 8 y \cdot 1 = \ln (x)$

Step 3.3.3

Multiply $- 8$ by $1$ .

$- 8 y = \ln (x)$

Step 3.4

Divide each term in $- 8 y = \ln (x)$ by $- 8$ and simplify.

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Step 3.4.1

Divide each term in $- 8 y = \ln (x)$ by $- 8$ .

$\frac{- 8 y}{- 8} = \frac{\ln (x)}{- 8}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $- 8$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{- 8 y}{- 8} = \frac{\ln (x)}{- 8}$

Step 3.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{\ln (x)}{- 8}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Move the negative in front of the fraction.

$y = - \frac{\ln (x)}{8}$

Step 4

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

$f^{- 1} (x) = - \frac{\ln (x)}{8}$

Step 5

Verify if $f^{- 1} (x) = - \frac{\ln (x)}{8}$ is the inverse of $f (x) = e^{- 8 x}$ .

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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 $f^{- 1} (e^{- 8 x})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (e^{- 8 x}) = - \frac{\ln (e^{- 8 x})}{8}$

Step 5.2.3

Expand $\ln (e^{- 8 x})$ by moving $- 8 x$ outside the logarithm.

$f^{- 1} (e^{- 8 x}) = - \frac{- 8 x \ln (e)}{8}$

Step 5.2.4

Cancel the common factor of $- 8$ and $8$ .

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Step 5.2.4.1

Factor $8$ out of $- 8 x \ln (e)$ .

$f^{- 1} (e^{- 8 x}) = - \frac{8 (- x \ln (e))}{8}$

Step 5.2.4.2

Cancel the common factors.

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Step 5.2.4.2.1

Factor $8$ out of $8$ .

$f^{- 1} (e^{- 8 x}) = - \frac{8 (- x \ln (e))}{8 (1)}$

Step 5.2.4.2.2

Cancel the common factor.

$f^{- 1} (e^{- 8 x}) = - \frac{8 (- x \ln (e))}{8 \cdot 1}$

Step 5.2.4.2.3

Rewrite the expression.

$f^{- 1} (e^{- 8 x}) = - \frac{- x \ln (e)}{1}$

Step 5.2.4.2.4

Divide $- x \ln (e)$ by $1$ .

$f^{- 1} (e^{- 8 x}) = - (- x \ln (e))$

Step 5.2.5

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

$f^{- 1} (e^{- 8 x}) = - (- x \cdot 1)$

Step 5.2.6

Multiply $- 1$ by $1$ .

$f^{- 1} (e^{- 8 x}) = x$

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 $f (- \frac{\ln (x)}{8})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- \frac{\ln (x)}{8}) = e^{- 8 (- \frac{\ln (x)}{8})}$

Step 5.3.3

Cancel the common factor of $8$ .

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Step 5.3.3.1

Move the leading negative in $- \frac{\ln (x)}{8}$ into the numerator.

$f (- \frac{\ln (x)}{8}) = e^{- 8 \frac{- \ln (x)}{8}}$

Step 5.3.3.2

Factor $8$ out of $- 8$ .

$f (- \frac{\ln (x)}{8}) = e^{8 (- 1) (\frac{- \ln (x)}{8})}$

Step 5.3.3.3

Cancel the common factor.

$f (- \frac{\ln (x)}{8}) = e^{8 \cdot (- 1 \frac{- \ln (x)}{8})}$

Step 5.3.3.4

Rewrite the expression.

$f (- \frac{\ln (x)}{8}) = e^{- 1 (- \ln (x))}$

Step 5.3.4

Multiply.

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Step 5.3.4.1

Multiply $- 1$ by $- 1$ .

$f (- \frac{\ln (x)}{8}) = e^{1 \ln (x)}$

Step 5.3.4.2

Multiply $\ln (x)$ by $1$ .

$f (- \frac{\ln (x)}{8}) = e^{\ln (x)}$

Step 5.3.5

Exponentiation and log are inverse functions.

$f (- \frac{\ln (x)}{8}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{\ln (x)}{8}$ is the inverse of $f (x) = e^{- 8 x}$ .

$f^{- 1} (x) = - \frac{\ln (x)}{8}$