Trigonometry Examples

$f (t) = 3 e^{0.1 t}$

Step 1

Write $f (t) = 3 e^{0.1 t}$ as an equation.

$y = 3 e^{0.1 t}$

Step 2

Interchange the variables.

$t = 3 e^{0.1 y}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $3 e^{0.1 y} = t$ .

$3 e^{0.1 y} = t$

Step 3.2

Divide each term in $3 e^{0.1 y} = t$ by $3$ and simplify.

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Step 3.2.1

Divide each term in $3 e^{0.1 y} = t$ by $3$ .

$\frac{3 e^{0.1 y}}{3} = \frac{t}{3}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $3$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{3 e^{0.1 y}}{3} = \frac{t}{3}$

Step 3.2.2.1.2

Divide $e^{0.1 y}$ by $1$ .

$e^{0.1 y} = \frac{t}{3}$

Step 3.3

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

$\ln (e^{0.1 y}) = \ln (\frac{t}{3})$

Step 3.4

Expand the left side.

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Step 3.4.1

Expand $\ln (e^{0.1 y})$ by moving $0.1 y$ outside the logarithm.

$0.1 y \ln (e) = \ln (\frac{t}{3})$

Step 3.4.2

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

$0.1 y \cdot 1 = \ln (\frac{t}{3})$

Step 3.4.3

Multiply $0.1$ by $1$ .

$0.1 y = \ln (\frac{t}{3})$

Step 3.5

Divide each term in $0.1 y = \ln (\frac{t}{3})$ by $0.1$ and simplify.

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Step 3.5.1

Divide each term in $0.1 y = \ln (\frac{t}{3})$ by $0.1$ .

$\frac{0.1 y}{0.1} = \frac{\ln (\frac{t}{3})}{0.1}$

Step 3.5.2

Simplify the left side.

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Step 3.5.2.1

Cancel the common factor of $0.1$ .

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Step 3.5.2.1.1

Cancel the common factor.

$\frac{0.1 y}{0.1} = \frac{\ln (\frac{t}{3})}{0.1}$

Step 3.5.2.1.2

Divide $y$ by $1$ .

$y = \frac{\ln (\frac{t}{3})}{0.1}$

Step 3.5.3

Simplify the right side.

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Step 3.5.3.1

Multiply by $1$ .

$y = \frac{1 \ln (\frac{t}{3})}{0.1}$

Step 3.5.3.2

Factor $0.1$ out of $0.1$ .

$y = \frac{1 \ln (\frac{t}{3})}{0.1 (1)}$

Step 3.5.3.3

Separate fractions.

$y = \frac{1}{0.1} \cdot \frac{\ln (\frac{t}{3})}{1}$

Step 3.5.3.4

Divide $1$ by $0.1$ .

$y = 10 \frac{\ln (\frac{t}{3})}{1}$

Step 3.5.3.5

Divide $\ln (\frac{t}{3})$ by $1$ .

$y = 10 \ln (\frac{t}{3})$

Step 4

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

$f^{- 1} (t) = 10 \ln (\frac{t}{3})$

Step 5

Verify if $f^{- 1} (t) = 10 \ln (\frac{t}{3})$ is the inverse of $f (t) = 3 e^{0.1 t}$ .

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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 $f^{- 1} (3 e^{0.1 t})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (3 e^{0.1 t}) = 10 \ln (\frac{3 e^{0.1 t}}{3})$

Step 5.2.3

Cancel the common factor of $3$ .

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Step 5.2.3.1

Cancel the common factor.

$f^{- 1} (3 e^{0.1 t}) = 10 \ln (\frac{3 e^{0.1 t}}{3})$

Step 5.2.3.2

Divide $e^{0.1 t}$ by $1$ .

$f^{- 1} (3 e^{0.1 t}) = 10 \ln (e^{0.1 t})$

Step 5.2.4

Use logarithm rules to move $0.1 t$ out of the exponent.

$f^{- 1} (3 e^{0.1 t}) = 10 (0.1 t \ln (e))$

Step 5.2.5

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

$f^{- 1} (3 e^{0.1 t}) = 10 (0.1 t \cdot 1)$

Step 5.2.6

Multiply $0.1$ by $1$ .

$f^{- 1} (3 e^{0.1 t}) = 10 (0.1 t)$

Step 5.2.7

Multiply $10 (0.1 t)$ .

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Step 5.2.7.1

Multiply $0.1$ by $10$ .

$f^{- 1} (3 e^{0.1 t}) = 1 t$

Step 5.2.7.2

Multiply $t$ by $1$ .

$f^{- 1} (3 e^{0.1 t}) = t$

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 $f (10 \ln (\frac{t}{3}))$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (10 \ln (\frac{t}{3})) = 3 e^{0.1 (10 \ln (\frac{t}{3}))}$

Step 5.3.3

Simplify $10 \ln (\frac{t}{3})$ by moving $10$ inside the logarithm.

$f (10 \ln (\frac{t}{3})) = 3 e^{0.1 \ln ({(\frac{t}{3})}^{10})}$

Step 5.3.4

Simplify $0.1 \ln ({(\frac{t}{3})}^{10})$ by moving $0.1$ inside the logarithm.

$f (10 \ln (\frac{t}{3})) = 3 e^{\ln ({({(\frac{t}{3})}^{10})}^{0.1})}$

Step 5.3.5

Exponentiation and log are inverse functions.

$f (10 \ln (\frac{t}{3})) = 3 {({(\frac{t}{3})}^{10})}^{0.1}$

Step 5.3.6

Multiply the exponents in ${({(\frac{t}{3})}^{10})}^{0.1}$ .

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Step 5.3.6.1

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

$f (10 \ln (\frac{t}{3})) = 3 {(\frac{t}{3})}^{10 \cdot 0.1}$

Step 5.3.6.2

Multiply $10$ by $0.1$ .

$f (10 \ln (\frac{t}{3})) = 3 (\frac{t}{3})$

Step 5.3.7

Cancel the common factor of $3$ .

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Step 5.3.7.1

Cancel the common factor.

$f (10 \ln (\frac{t}{3})) = 3 (\frac{t}{3})$

Step 5.3.7.2

Rewrite the expression.

$f (10 \ln (\frac{t}{3})) = t$

Step 5.4

Since $f^{- 1} (f (t)) = t$ and $f (f^{- 1} (t)) = t$ , then $f^{- 1} (t) = 10 \ln (\frac{t}{3})$ is the inverse of $f (t) = 3 e^{0.1 t}$ .

$f^{- 1} (t) = 10 \ln (\frac{t}{3})$