Algebra Examples

$y = e^{8 - x}$

Step 1

Interchange the variables.

$x = e^{8 - y}$

Step 2

Solve for $y$ .

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Step 2.1

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

$e^{8 - y} = x$

Step 2.2

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

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

Step 2.3

Expand the left side.

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

Multiply $8 - y$ by $1$ .

$8 - y = \ln (x)$

Step 2.4

Subtract $8$ from both sides of the equation.

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

Step 2.5

Divide each term in $- y = \ln (x) - 8$ by $- 1$ and simplify.

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Step 2.5.1

Divide each term in $- y = \ln (x) - 8$ by $- 1$ .

$\frac{- y}{- 1} = \frac{\ln (x)}{- 1} + \frac{- 8}{- 1}$

Step 2.5.2

Simplify the left side.

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Step 2.5.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{\ln (x)}{- 1} + \frac{- 8}{- 1}$

Step 2.5.2.2

Divide $y$ by $1$ .

$y = \frac{\ln (x)}{- 1} + \frac{- 8}{- 1}$

Step 2.5.3

Simplify the right side.

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Step 2.5.3.1

Simplify each term.

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Step 2.5.3.1.1

Move the negative one from the denominator of $\frac{\ln (x)}{- 1}$ .

$y = - 1 \cdot \ln (x) + \frac{- 8}{- 1}$

Step 2.5.3.1.2

Rewrite $- 1 \cdot \ln (x)$ as $- \ln (x)$ .

$y = - \ln (x) + \frac{- 8}{- 1}$

Step 2.5.3.1.3

Divide $- 8$ by $- 1$ .

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

Step 3

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

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

Step 4

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

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Step 4.1

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

Step 4.2

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

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Step 4.2.1

Set up the composite result function.

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

Step 4.2.2

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

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

Step 4.2.3

Simplify each term.

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Step 4.2.3.1

Use logarithm rules to move $8 - x$ out of the exponent.

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

Step 4.2.3.2

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

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

Step 4.2.3.3

Multiply $8 - x$ by $1$ .

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

Step 4.2.3.4

Apply the distributive property.

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

Step 4.2.3.5

Multiply $- 1$ by $8$ .

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

Step 4.2.3.6

Multiply $- - x$ .

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Step 4.2.3.6.1

Multiply $- 1$ by $- 1$ .

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

Step 4.2.3.6.2

Multiply $x$ by $1$ .

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

Step 4.2.4

Combine the opposite terms in $- 8 + x + 8$ .

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Step 4.2.4.1

Add $- 8$ and $8$ .

$f^{- 1} (e^{8 - x}) = x + 0$

Step 4.2.4.2

Add $x$ and $0$ .

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

Step 4.3

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

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Step 4.3.1

Set up the composite result function.

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

Step 4.3.2

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

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

Step 4.3.3

Simplify each term.

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Step 4.3.3.1

Apply the distributive property.

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

Step 4.3.3.2

Multiply $- - \ln (x)$ .

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Step 4.3.3.2.1

Multiply $- 1$ by $- 1$ .

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

Step 4.3.3.2.2

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

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

Step 4.3.3.3

Multiply $- 1$ by $8$ .

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

Step 4.3.4

Combine the opposite terms in $8 + \ln (x) - 8$ .

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Step 4.3.4.1

Subtract $8$ from $8$ .

$f (- \ln (x) + 8) = e^{0 + \ln (x)}$

Step 4.3.4.2

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

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

Step 4.3.5

Exponentiation and log are inverse functions.

$f (- \ln (x) + 8) = x$

Step 4.4

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

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