Calculus Examples

$f (x) = \sqrt{3 - e^{2 x}}$

Step 1

Write $f (x) = \sqrt{3 - e^{2 x}}$ as an equation.

$y = \sqrt{3 - e^{2 x}}$

Step 2

Interchange the variables.

$x = \sqrt{3 - e^{2 y}}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\sqrt{3 - e^{2 y}} = x$ .

$\sqrt{3 - e^{2 y}} = x$

Step 3.2

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{3 - e^{2 y}}}^{2} = x^{2}$

Step 3.3

Simplify each side of the equation.

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Step 3.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 - e^{2 y}}$ as ${(3 - e^{2 y})}^{\frac{1}{2}}$ .

${({(3 - e^{2 y})}^{\frac{1}{2}})}^{2} = x^{2}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Simplify ${({(3 - e^{2 y})}^{\frac{1}{2}})}^{2}$ .

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Step 3.3.2.1.1

Multiply the exponents in ${({(3 - e^{2 y})}^{\frac{1}{2}})}^{2}$ .

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Step 3.3.2.1.1.1

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

${(3 - e^{2 y})}^{\frac{1}{2} \cdot 2} = x^{2}$

Step 3.3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.3.2.1.1.2.1

Cancel the common factor.

${(3 - e^{2 y})}^{\frac{1}{2} \cdot 2} = x^{2}$

Step 3.3.2.1.1.2.2

Rewrite the expression.

${(3 - e^{2 y})}^{1} = x^{2}$

Step 3.3.2.1.2

Simplify.

$3 - e^{2 y} = x^{2}$

Step 3.4

Solve for $y$ .

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Step 3.4.1

Subtract $3$ from both sides of the equation.

$- e^{2 y} = x^{2} - 3$

Step 3.4.2

Divide each term in $- e^{2 y} = x^{2} - 3$ by $- 1$ and simplify.

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Step 3.4.2.1

Divide each term in $- e^{2 y} = x^{2} - 3$ by $- 1$ .

$\frac{- e^{2 y}}{- 1} = \frac{x^{2}}{- 1} + \frac{- 3}{- 1}$

Step 3.4.2.2

Simplify the left side.

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Step 3.4.2.2.1

Dividing two negative values results in a positive value.

$\frac{e^{2 y}}{1} = \frac{x^{2}}{- 1} + \frac{- 3}{- 1}$

Step 3.4.2.2.2

Divide $e^{2 y}$ by $1$ .

$e^{2 y} = \frac{x^{2}}{- 1} + \frac{- 3}{- 1}$

Step 3.4.2.3

Simplify the right side.

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Step 3.4.2.3.1

Simplify each term.

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Step 3.4.2.3.1.1

Move the negative one from the denominator of $\frac{x^{2}}{- 1}$ .

$e^{2 y} = - 1 \cdot x^{2} + \frac{- 3}{- 1}$

Step 3.4.2.3.1.2

Rewrite $- 1 \cdot x^{2}$ as $- x^{2}$ .

$e^{2 y} = - x^{2} + \frac{- 3}{- 1}$

Step 3.4.2.3.1.3

Divide $- 3$ by $- 1$ .

$e^{2 y} = - x^{2} + 3$

Step 3.4.3

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

$\ln (e^{2 y}) = \ln (- x^{2} + 3)$

Step 3.4.4

Expand the left side.

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Step 3.4.4.1

Expand $\ln (e^{2 y})$ by moving $2 y$ outside the logarithm.

$2 y \ln (e) = \ln (- x^{2} + 3)$

Step 3.4.4.2

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

$2 y \cdot 1 = \ln (- x^{2} + 3)$

Step 3.4.4.3

Multiply $2$ by $1$ .

$2 y = \ln (- x^{2} + 3)$

Step 3.4.5

Divide each term in $2 y = \ln (- x^{2} + 3)$ by $2$ and simplify.

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Step 3.4.5.1

Divide each term in $2 y = \ln (- x^{2} + 3)$ by $2$ .

$\frac{2 y}{2} = \frac{\ln (- x^{2} + 3)}{2}$

Step 3.4.5.2

Simplify the left side.

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Step 3.4.5.2.1

Cancel the common factor of $2$ .

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Step 3.4.5.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{\ln (- x^{2} + 3)}{2}$

Step 3.4.5.2.1.2

Divide $y$ by $1$ .

$y = \frac{\ln (- x^{2} + 3)}{2}$

Step 4

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

$f^{- 1} (x) = \frac{\ln (- x^{2} + 3)}{2}$

Step 5

Verify if $f^{- 1} (x) = \frac{\ln (- x^{2} + 3)}{2}$ is the inverse of $f (x) = \sqrt{3 - e^{2 x}}$ .

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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 $f^{- 1} (\sqrt{3 - e^{2 x}})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\sqrt{3 - e^{2 x}}) = \frac{\ln (- {(\sqrt{3 - e^{2 x}})}^{2} + 3)}{2}$

Step 5.2.3

Simplify $\frac{1}{2} \ln (- {\sqrt{3 - e^{2 x}}}^{2} + 3)$ by moving $\frac{1}{2}$ inside the logarithm.

$f^{- 1} (\sqrt{3 - e^{2 x}}) = \ln ({(- {\sqrt{3 - e^{2 x}}}^{2} + 3)}^{\frac{1}{2}})$

Step 5.2.4

Simplify each term.

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Step 5.2.4.1

Rewrite ${\sqrt{3 - e^{2 x}}}^{2}$ as $3 - e^{2 x}$ .

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Step 5.2.4.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 - e^{2 x}}$ as ${(3 - e^{2 x})}^{\frac{1}{2}}$ .

$f^{- 1} (\sqrt{3 - e^{2 x}}) = \ln ({(- {({(3 - e^{2 x})}^{\frac{1}{2}})}^{2} + 3)}^{\frac{1}{2}})$

Step 5.2.4.1.2

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

$f^{- 1} (\sqrt{3 - e^{2 x}}) = \ln ({(- {(3 - e^{2 x})}^{\frac{1}{2} \cdot 2} + 3)}^{\frac{1}{2}})$

Step 5.2.4.1.3

Combine $\frac{1}{2}$ and $2$ .

$f^{- 1} (\sqrt{3 - e^{2 x}}) = \ln ({(- {(3 - e^{2 x})}^{\frac{2}{2}} + 3)}^{\frac{1}{2}})$

Step 5.2.4.1.4

Cancel the common factor of $2$ .

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Step 5.2.4.1.4.1

Cancel the common factor.

$f^{- 1} (\sqrt{3 - e^{2 x}}) = \ln ({(- {(3 - e^{2 x})}^{\frac{2}{2}} + 3)}^{\frac{1}{2}})$

Step 5.2.4.1.4.2

Rewrite the expression.

$f^{- 1} (\sqrt{3 - e^{2 x}}) = \ln ({(- (3 - e^{2 x}) + 3)}^{\frac{1}{2}})$

Step 5.2.4.1.5

Simplify.

$f^{- 1} (\sqrt{3 - e^{2 x}}) = \ln ({(- (3 - e^{2 x}) + 3)}^{\frac{1}{2}})$

Step 5.2.4.2

Apply the distributive property.

$f^{- 1} (\sqrt{3 - e^{2 x}}) = \ln ({(- 1 \cdot 3 + e^{2 x} + 3)}^{\frac{1}{2}})$

Step 5.2.4.3

Multiply $- 1$ by $3$ .

$f^{- 1} (\sqrt{3 - e^{2 x}}) = \ln ({(- 3 + e^{2 x} + 3)}^{\frac{1}{2}})$

Step 5.2.4.4

Multiply $- - e^{2 x}$ .

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Step 5.2.4.4.1

Multiply $- 1$ by $- 1$ .

$f^{- 1} (\sqrt{3 - e^{2 x}}) = \ln ({(- 3 + 1 e^{2 x} + 3)}^{\frac{1}{2}})$

Step 5.2.4.4.2

Multiply $e^{2 x}$ by $1$ .

$f^{- 1} (\sqrt{3 - e^{2 x}}) = \ln ({(- 3 + e^{2 x} + 3)}^{\frac{1}{2}})$

Step 5.2.5

Simplify by adding terms.

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Step 5.2.5.1

Combine the opposite terms in $- 3 + e^{2 x} + 3$ .

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Step 5.2.5.1.1

Add $- 3$ and $3$ .

$f^{- 1} (\sqrt{3 - e^{2 x}}) = \ln ({(0 + e^{2 x})}^{\frac{1}{2}})$

Step 5.2.5.1.2

Add $0$ and $e^{2 x}$ .

$f^{- 1} (\sqrt{3 - e^{2 x}}) = \ln ({(e^{2 x})}^{\frac{1}{2}})$

Step 5.2.5.2

Multiply the exponents in ${(e^{2 x})}^{\frac{1}{2}}$ .

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Step 5.2.5.2.1

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

$f^{- 1} (\sqrt{3 - e^{2 x}}) = \ln (e^{2 x (\frac{1}{2})})$

Step 5.2.5.2.2

Cancel the common factor of $2$ .

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Step 5.2.5.2.2.1

Factor $2$ out of $2 x$ .

$f^{- 1} (\sqrt{3 - e^{2 x}}) = \ln (e^{2 (x) (\frac{1}{2})})$

Step 5.2.5.2.2.2

Cancel the common factor.

$f^{- 1} (\sqrt{3 - e^{2 x}}) = \ln (e^{2 x (\frac{1}{2})})$

Step 5.2.5.2.2.3

Rewrite the expression.

$f^{- 1} (\sqrt{3 - e^{2 x}}) = \ln (e^{x})$

Step 5.2.6

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

$f^{- 1} (\sqrt{3 - e^{2 x}}) = x \ln (e)$

Step 5.2.7

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

$f^{- 1} (\sqrt{3 - e^{2 x}}) = x \cdot 1$

Step 5.2.8

Multiply $x$ by $1$ .

$f^{- 1} (\sqrt{3 - e^{2 x}}) = x$

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 $f (\frac{\ln (- x^{2} + 3)}{2})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{\ln (- x^{2} + 3)}{2}) = \sqrt{3 - e^{2 (\frac{\ln (- x^{2} + 3)}{2})}}$

Step 5.3.3

Cancel the common factor of $2$ .

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Step 5.3.3.1

Cancel the common factor.

$f (\frac{\ln (- x^{2} + 3)}{2}) = \sqrt{3 - e^{2 (\frac{\ln (- x^{2} + 3)}{2})}}$

Step 5.3.3.2

Rewrite the expression.

$f (\frac{\ln (- x^{2} + 3)}{2}) = \sqrt{3 - e^{\ln (- x^{2} + 3)}}$

Step 5.3.4

Exponentiation and log are inverse functions.

$f (\frac{\ln (- x^{2} + 3)}{2}) = \sqrt{3 - (- x^{2} + 3)}$

Step 5.3.5

Apply the distributive property.

$f (\frac{\ln (- x^{2} + 3)}{2}) = \sqrt{3 + x^{2} - 1 \cdot 3}$

Step 5.3.6

Multiply $- 1$ by $3$ .

$f (\frac{\ln (- x^{2} + 3)}{2}) = \sqrt{3 + x^{2} - 3}$

Step 5.3.7

Subtract $3$ from $3$ .

$f (\frac{\ln (- x^{2} + 3)}{2}) = \sqrt{x^{2} + 0}$

Step 5.3.8

Add $x^{2}$ and $0$ .

$f (\frac{\ln (- x^{2} + 3)}{2}) = \sqrt{x^{2}}$

Step 5.3.9

Pull terms out from under the radical, assuming positive real numbers.

$f (\frac{\ln (- x^{2} + 3)}{2}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{\ln (- x^{2} + 3)}{2}$ is the inverse of $f (x) = \sqrt{3 - e^{2 x}}$ .

$f^{- 1} (x) = \frac{\ln (- x^{2} + 3)}{2}$