Finite Math Examples

$f (x) = \sin (\sqrt{e^{x} + 1})$

Step 1

Write $f (x) = \sin (\sqrt{e^{x} + 1})$ as an equation.

$y = \sin (\sqrt{e^{x} + 1})$

Step 2

Interchange the variables.

$x = \sin (\sqrt{e^{y} + 1})$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\sin (\sqrt{e^{y} + 1}) = x$ .

$\sin (\sqrt{e^{y} + 1}) = x$

Step 3.2

Substitute $u$ for $\sqrt{e^{y} + 1}$ .

$\sin (u) = x$

Step 3.3

Take the inverse sine of both sides of the equation to extract $u$ from inside the sine.

$u = \arcsin (x)$

Step 3.4

Substitute $\sqrt{e^{y} + 1}$ for $u$ and solve $\sqrt{e^{y} + 1} = \arcsin (x)$

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Step 3.4.1

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

${\sqrt{e^{y} + 1}}^{2} = {\arcsin (x)}^{2}$

Step 3.4.2

Simplify each side of the equation.

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Step 3.4.2.1

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

${({(e^{y} + 1)}^{\frac{1}{2}})}^{2} = {\arcsin (x)}^{2}$

Step 3.4.2.2

Simplify the left side.

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Step 3.4.2.2.1

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

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Step 3.4.2.2.1.1

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

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Step 3.4.2.2.1.1.1

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

${(e^{y} + 1)}^{\frac{1}{2} \cdot 2} = {\arcsin (x)}^{2}$

Step 3.4.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.4.2.2.1.1.2.1

Cancel the common factor.

${(e^{y} + 1)}^{\frac{1}{2} \cdot 2} = {\arcsin (x)}^{2}$

Step 3.4.2.2.1.1.2.2

Rewrite the expression.

${(e^{y} + 1)}^{1} = {\arcsin (x)}^{2}$

Step 3.4.2.2.1.2

Simplify.

$e^{y} + 1 = {\arcsin (x)}^{2}$

Step 3.4.3

Solve for $y$ .

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Step 3.4.3.1

Subtract $1$ from both sides of the equation.

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

Step 3.4.3.2

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

$\ln (e^{y}) = \ln ({\arcsin (x)}^{2} - 1)$

Step 3.4.3.3

Expand the left side.

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Step 3.4.3.3.1

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

$y \ln (e) = \ln ({\arcsin (x)}^{2} - 1)$

Step 3.4.3.3.2

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

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

Step 3.4.3.3.3

Multiply $y$ by $1$ .

$y = \ln ({\arcsin (x)}^{2} - 1)$

Step 4

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

$f^{- 1} (x) = \ln ({\arcsin (x)}^{2} - 1)$

Step 5

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

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

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

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

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

Step 5.3.3

Exponentiation and log are inverse functions.

$f (\ln ({\arcsin (x)}^{2} - 1)) = \sin (\sqrt{{\arcsin (x)}^{2} - 1 + 1})$

Step 5.3.4

Add $- 1$ and $1$ .

$f (\ln ({\arcsin (x)}^{2} - 1)) = \sin (\sqrt{{\arcsin (x)}^{2} + 0})$

Step 5.3.5

Add ${\arcsin (x)}^{2}$ and $0$ .

$f (\ln ({\arcsin (x)}^{2} - 1)) = \sin (\sqrt{{\arcsin (x)}^{2}})$

Step 5.3.6

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

$f (\ln ({\arcsin (x)}^{2} - 1)) = \sin (\arcsin (x))$

Step 5.3.7

The functions sine and arcsine are inverses.

$f (\ln ({\arcsin (x)}^{2} - 1)) = x$

Step 5.4

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

$f^{- 1} (x) = \ln ({\arcsin (x)}^{2} - 1)$