Precalculus Examples

$p (x) = 37.3 e^{0.017 x}$

Step 1

Write $p (x) = 37.3 e^{0.017 x}$ as an equation.

$y = 37.3 e^{0.017 x}$

Step 2

Interchange the variables.

$x = 37.3 e^{0.017 y}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $37.3 e^{0.017 y} = x$ .

$37.3 e^{0.017 y} = x$

Step 3.2

Divide each term in $37.3 e^{0.017 y} = x$ by $37.3$ and simplify.

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Step 3.2.1

Divide each term in $37.3 e^{0.017 y} = x$ by $37.3$ .

$\frac{37.3 e^{0.017 y}}{37.3} = \frac{x}{37.3}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $37.3$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{37.3 e^{0.017 y}}{37.3} = \frac{x}{37.3}$

Step 3.2.2.1.2

Divide $e^{0.017 y}$ by $1$ .

$e^{0.017 y} = \frac{x}{37.3}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Multiply by $1$ .

$e^{0.017 y} = \frac{1 x}{37.3}$

Step 3.2.3.2

Factor $37.3$ out of $37.3$ .

$e^{0.017 y} = \frac{1 x}{37.3 (1)}$

Step 3.2.3.3

Separate fractions.

$e^{0.017 y} = \frac{1}{37.3} \cdot \frac{x}{1}$

Step 3.2.3.4

Divide $1$ by $37.3$ .

$e^{0.017 y} = 0.02680965 \frac{x}{1}$

Step 3.2.3.5

Divide $x$ by $1$ .

$e^{0.017 y} = 0.02680965 x$

Step 3.3

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

$\ln (e^{0.017 y}) = \ln (0.02680965 x)$

Step 3.4

Expand the left side.

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Step 3.4.1

Expand $\ln (e^{0.017 y})$ by moving $0.017 y$ outside the logarithm.

$0.017 y \ln (e) = \ln (0.02680965 x)$

Step 3.4.2

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

$0.017 y \cdot 1 = \ln (0.02680965 x)$

Step 3.4.3

Multiply $0.017$ by $1$ .

$0.017 y = \ln (0.02680965 x)$

Step 3.5

Divide each term in $0.017 y = \ln (0.02680965 x)$ by $0.017$ and simplify.

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Step 3.5.1

Divide each term in $0.017 y = \ln (0.02680965 x)$ by $0.017$ .

$\frac{0.017 y}{0.017} = \frac{\ln (0.02680965 x)}{0.017}$

Step 3.5.2

Simplify the left side.

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Step 3.5.2.1

Cancel the common factor of $0.017$ .

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Step 3.5.2.1.1

Cancel the common factor.

$\frac{0.017 y}{0.017} = \frac{\ln (0.02680965 x)}{0.017}$

Step 3.5.2.1.2

Divide $y$ by $1$ .

$y = \frac{\ln (0.02680965 x)}{0.017}$

Step 3.5.3

Simplify the right side.

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Step 3.5.3.1

Multiply by $1$ .

$y = \frac{1 \ln (0.02680965 x)}{0.017}$

Step 3.5.3.2

Factor $0.017$ out of $0.017$ .

$y = \frac{1 \ln (0.02680965 x)}{0.017 (1)}$

Step 3.5.3.3

Separate fractions.

$y = \frac{1}{0.017} \cdot \frac{\ln (0.02680965 x)}{1}$

Step 3.5.3.4

Divide $1$ by $0.017$ .

$y = 58.82352941 \frac{\ln (0.02680965 x)}{1}$

Step 3.5.3.5

Divide $\ln (0.02680965 x)$ by $1$ .

$y = 58.82352941 \ln (0.02680965 x)$

Step 4

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

$p^{- 1} (x) = 58.82352941 \ln (0.02680965 x)$

Step 5

Verify if $p^{- 1} (x) = 58.82352941 \ln (0.02680965 x)$ is the inverse of $p (x) = 37.3 e^{0.017 x}$ .

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Step 5.1

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

Step 5.2

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

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Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

Evaluate $p^{- 1} (37.3 e^{0.017 x})$ by substituting in the value of $p$ into $p^{- 1}$ .

$p^{- 1} (37.3 e^{0.017 x}) = 58.82352941 \ln (0.02680965 (37.3 e^{0.017 x}))$

Step 5.2.3

Multiply $37.3$ by $0.02680965$ .

$p^{- 1} (37.3 e^{0.017 x}) = 58.82352941 \ln (1 e^{0.017 x})$

Step 5.2.4

Rewrite $\ln (1 e^{0.017 x})$ as $\ln (1) + \ln (e^{0.017 x})$ .

$p^{- 1} (37.3 e^{0.017 x}) = 58.82352941 (\ln (1) + \ln (e^{0.017 x}))$

Step 5.2.5

Use logarithm rules to move $0.017 x$ out of the exponent.

$p^{- 1} (37.3 e^{0.017 x}) = 58.82352941 (\ln (1) + 0.017 x \ln (e))$

Step 5.2.6

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

$p^{- 1} (37.3 e^{0.017 x}) = 58.82352941 (\ln (1) + 0.017 x \cdot 1)$

Step 5.2.7

Multiply $0.017$ by $1$ .

$p^{- 1} (37.3 e^{0.017 x}) = 58.82352941 (\ln (1) + 0.017 x)$

Step 5.2.8

Apply the distributive property.

$p^{- 1} (37.3 e^{0.017 x}) = 58.82352941 \ln (1) + 58.82352941 (0.017 x)$

Step 5.2.9

Simplify $58.82352941 \ln (1)$ by moving $58.82352941$ inside the logarithm.

$p^{- 1} (37.3 e^{0.017 x}) = \ln (1^{58.82352941}) + 58.82352941 (0.017 x)$

Step 5.2.10

Multiply $58.82352941 (0.017 x)$ .

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Step 5.2.10.1

Multiply $0.017$ by $58.82352941$ .

$p^{- 1} (37.3 e^{0.017 x}) = \ln (1^{58.82352941}) + 1 x$

Step 5.2.10.2

Multiply $x$ by $1$ .

$p^{- 1} (37.3 e^{0.017 x}) = \ln (1^{58.82352941}) + x$

Step 5.2.11

Raise $1$ to the power of $58.82352941$ .

$p^{- 1} (37.3 e^{0.017 x}) = \ln (1) + x$

Step 5.2.12

Reorder $\ln (1)$ and $x$ .

$p^{- 1} (37.3 e^{0.017 x}) = x + \ln (1)$

Step 5.3

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

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Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

Evaluate $p (58.82352941 \ln (0.02680965 x))$ by substituting in the value of $p^{- 1}$ into $p$ .

$p (58.82352941 \ln (0.02680965 x)) = 37.3 e^{0.017 (58.82352941 \ln (0.02680965 x))}$

Step 5.3.3

Simplify $58.82352941 \ln (0.02680965 x)$ by moving $58.82352941$ inside the logarithm.

$p (58.82352941 \ln (0.02680965 x)) = 37.3 e^{0.017 \ln ({(0.02680965 x)}^{58.82352941})}$

Step 5.3.4

Apply the product rule to $0.02680965 x$ .

$p (58.82352941 \ln (0.02680965 x)) = 37.3 e^{0.017 \ln ({0.02680965}^{58.82352941} x^{58.82352941})}$

Step 5.3.5

Raise $0.02680965$ to the power of $58.82352941$ .

$p (58.82352941 \ln (0.02680965 x)) = 37.3 e^{0.017 \ln (0 x^{58.82352941})}$

Step 5.3.6

Multiply $0$ by $x^{58.82352941}$ .

$p (58.82352941 \ln (0.02680965 x)) = 37.3 e^{0.017 \ln (0)}$

Step 5.3.7

The natural logarithm of zero is undefined.

$p (58.82352941 \ln (0.02680965 x)) = Undefined$

Step 5.4

Since $p^{- 1} (p (x)) = x$ and $p (p^{- 1} (x)) = x$ , then $p^{- 1} (x) = 58.82352941 \ln (0.02680965 x)$ is the inverse of $p (x) = 37.3 e^{0.017 x}$ .

$p^{- 1} (x) = 58.82352941 \ln (0.02680965 x)$