Precalculus Examples

$f (t) = 26 \cdot e^{0.5 t}$

Step 1

Write $f (t) = 26 \cdot e^{0.5 t}$ as an equation.

$y = 26 \cdot e^{0.5 t}$

Step 2

Interchange the variables.

$t = 26 \cdot e^{0.5 y}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $26 \cdot e^{0.5 y} = t$ .

$26 \cdot e^{0.5 y} = t$

Step 3.2

Divide each term in $26 \cdot e^{0.5 y} = t$ by $26$ and simplify.

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Step 3.2.1

Divide each term in $26 \cdot e^{0.5 y} = t$ by $26$ .

$\frac{26 \cdot e^{0.5 y}}{26} = \frac{t}{26}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $26$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{26 \cdot e^{0.5 y}}{26} = \frac{t}{26}$

Step 3.2.2.1.2

Divide $e^{0.5 y}$ by $1$ .

$e^{0.5 y} = \frac{t}{26}$

Step 3.3

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

$\ln (e^{0.5 y}) = \ln (\frac{t}{26})$

Step 3.4

Expand the left side.

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Step 3.4.1

Expand $\ln (e^{0.5 y})$ by moving $0.5 y$ outside the logarithm.

$0.5 y \ln (e) = \ln (\frac{t}{26})$

Step 3.4.2

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

$0.5 y \cdot 1 = \ln (\frac{t}{26})$

Step 3.4.3

Multiply $0.5$ by $1$ .

$0.5 y = \ln (\frac{t}{26})$

Step 3.5

Divide each term in $0.5 y = \ln (\frac{t}{26})$ by $0.5$ and simplify.

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Step 3.5.1

Divide each term in $0.5 y = \ln (\frac{t}{26})$ by $0.5$ .

$\frac{0.5 y}{0.5} = \frac{\ln (\frac{t}{26})}{0.5}$

Step 3.5.2

Simplify the left side.

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Step 3.5.2.1

Cancel the common factor of $0.5$ .

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Step 3.5.2.1.1

Cancel the common factor.

$\frac{0.5 y}{0.5} = \frac{\ln (\frac{t}{26})}{0.5}$

Step 3.5.2.1.2

Divide $y$ by $1$ .

$y = \frac{\ln (\frac{t}{26})}{0.5}$

Step 3.5.3

Simplify the right side.

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Step 3.5.3.1

Multiply by $1$ .

$y = \frac{1 \ln (\frac{t}{26})}{0.5}$

Step 3.5.3.2

Factor $0.5$ out of $0.5$ .

$y = \frac{1 \ln (\frac{t}{26})}{0.5 (1)}$

Step 3.5.3.3

Separate fractions.

$y = \frac{1}{0.5} \cdot \frac{\ln (\frac{t}{26})}{1}$

Step 3.5.3.4

Divide $1$ by $0.5$ .

$y = 2 \frac{\ln (\frac{t}{26})}{1}$

Step 3.5.3.5

Divide $\ln (\frac{t}{26})$ by $1$ .

$y = 2 \ln (\frac{t}{26})$

Step 4

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

$f^{- 1} (t) = 2 \ln (\frac{t}{26})$

Step 5

Verify if $f^{- 1} (t) = 2 \ln (\frac{t}{26})$ is the inverse of $f (t) = 26 \cdot e^{0.5 t}$ .

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Step 5.1

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

Step 5.2

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

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Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

Evaluate $f^{- 1} (26 \cdot e^{0.5 t})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (26 \cdot e^{0.5 t}) = 2 \ln (\frac{26 \cdot e^{0.5 t}}{26})$

Step 5.2.3

Cancel the common factor of $26$ .

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Step 5.2.3.1

Cancel the common factor.

$f^{- 1} (26 \cdot e^{0.5 t}) = 2 \ln (\frac{26 \cdot e^{0.5 t}}{26})$

Step 5.2.3.2

Divide $e^{0.5 t}$ by $1$ .

$f^{- 1} (26 \cdot e^{0.5 t}) = 2 \ln (e^{0.5 t})$

Step 5.2.4

Use logarithm rules to move $0.5 t$ out of the exponent.

$f^{- 1} (26 \cdot e^{0.5 t}) = 2 (0.5 t \ln (e))$

Step 5.2.5

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

$f^{- 1} (26 \cdot e^{0.5 t}) = 2 (0.5 t \cdot 1)$

Step 5.2.6

Multiply $0.5$ by $1$ .

$f^{- 1} (26 \cdot e^{0.5 t}) = 2 (0.5 t)$

Step 5.2.7

Multiply $2 (0.5 t)$ .

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Step 5.2.7.1

Multiply $0.5$ by $2$ .

$f^{- 1} (26 \cdot e^{0.5 t}) = 1 t$

Step 5.2.7.2

Multiply $t$ by $1$ .

$f^{- 1} (26 \cdot e^{0.5 t}) = t$

Step 5.3

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

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Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

Evaluate $f (2 \ln (\frac{t}{26}))$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (2 \ln (\frac{t}{26})) = 26 \cdot e^{0.5 (2 \ln (\frac{t}{26}))}$

Step 5.3.3

Simplify $2 \ln (\frac{t}{26})$ by moving $2$ inside the logarithm.

$f (2 \ln (\frac{t}{26})) = 26 \cdot e^{0.5 \ln ({(\frac{t}{26})}^{2})}$

Step 5.3.4

Simplify $0.5 \ln ({(\frac{t}{26})}^{2})$ by moving $0.5$ inside the logarithm.

$f (2 \ln (\frac{t}{26})) = 26 \cdot e^{\ln ({({(\frac{t}{26})}^{2})}^{0.5})}$

Step 5.3.5

Exponentiation and log are inverse functions.

$f (2 \ln (\frac{t}{26})) = 26 \cdot {({(\frac{t}{26})}^{2})}^{0.5}$

Step 5.3.6

Multiply the exponents in ${({(\frac{t}{26})}^{2})}^{0.5}$ .

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Step 5.3.6.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (2 \ln (\frac{t}{26})) = 26 \cdot {(\frac{t}{26})}^{2 \cdot 0.5}$

Step 5.3.6.2

Multiply $2$ by $0.5$ .

$f (2 \ln (\frac{t}{26})) = 26 \cdot (\frac{t}{26})$

Step 5.3.7

Simplify.

$f (2 \ln (\frac{t}{26})) = 26 \cdot \frac{t}{26}$

Step 5.3.8

Cancel the common factor of $26$ .

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Step 5.3.8.1

Cancel the common factor.

$f (2 \ln (\frac{t}{26})) = 26 \cdot \frac{t}{26}$

Step 5.3.8.2

Rewrite the expression.

$f (2 \ln (\frac{t}{26})) = t$

Step 5.4

Since $f^{- 1} (f (t)) = t$ and $f (f^{- 1} (t)) = t$ , then $f^{- 1} (t) = 2 \ln (\frac{t}{26})$ is the inverse of $f (t) = 26 \cdot e^{0.5 t}$ .

$f^{- 1} (t) = 2 \ln (\frac{t}{26})$