Calculus Examples

$f (x) = \frac{1}{2} \cdot \sin (\frac{x}{2}) + \frac{1}{2}$

Step 1

Write $f (x) = \frac{1}{2} \cdot \sin (\frac{x}{2}) + \frac{1}{2}$ as an equation.

$y = \frac{1}{2} \cdot \sin (\frac{x}{2}) + \frac{1}{2}$

Step 2

Interchange the variables.

$x = \frac{1}{2} \cdot \sin (\frac{y}{2}) + \frac{1}{2}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{1}{2} \cdot \sin (\frac{y}{2}) + \frac{1}{2} = x$ .

$\frac{1}{2} \cdot \sin (\frac{y}{2}) + \frac{1}{2} = x$

Step 3.2

Combine $\frac{1}{2}$ and $\sin (\frac{y}{2})$ .

$\frac{\sin (\frac{y}{2})}{2} + \frac{1}{2} = x$

Step 3.3

Subtract $\frac{1}{2}$ from both sides of the equation.

$\frac{\sin (\frac{y}{2})}{2} = x - \frac{1}{2}$

Step 3.4

Multiply both sides of the equation by $2$ .

$2 \frac{\sin (\frac{y}{2})}{2} = 2 (x - \frac{1}{2})$

Step 3.5

Simplify both sides of the equation.

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Step 3.5.1

Simplify the left side.

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Step 3.5.1.1

Cancel the common factor of $2$ .

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Step 3.5.1.1.1

Cancel the common factor.

$2 \frac{\sin (\frac{y}{2})}{2} = 2 (x - \frac{1}{2})$

Step 3.5.1.1.2

Rewrite the expression.

$\sin (\frac{y}{2}) = 2 (x - \frac{1}{2})$

Step 3.5.2

Simplify the right side.

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Step 3.5.2.1

Simplify $2 (x - \frac{1}{2})$ .

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Step 3.5.2.1.1

Apply the distributive property.

$\sin (\frac{y}{2}) = 2 x + 2 (- \frac{1}{2})$

Step 3.5.2.1.2

Cancel the common factor of $2$ .

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Step 3.5.2.1.2.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$\sin (\frac{y}{2}) = 2 x + 2 (\frac{- 1}{2})$

Step 3.5.2.1.2.2

Cancel the common factor.

$\sin (\frac{y}{2}) = 2 x + 2 (\frac{- 1}{2})$

Step 3.5.2.1.2.3

Rewrite the expression.

$\sin (\frac{y}{2}) = 2 x - 1$

Step 3.6

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

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

Step 3.7

Multiply both sides of the equation by $2$ .

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

Step 3.8

Simplify the left side.

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Step 3.8.1

Cancel the common factor of $2$ .

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Step 3.8.1.1

Cancel the common factor.

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

Step 3.8.1.2

Rewrite the expression.

$y = 2 \arcsin (2 x - 1)$

Step 4

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

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

Step 5

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

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

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

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Combine $\frac{1}{2}$ and $\sin (\frac{x}{2})$ .

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

Step 5.2.3.2

Apply the distributive property.

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

Step 5.2.3.3

Cancel the common factor of $2$ .

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Step 5.2.3.3.1

Cancel the common factor.

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

Step 5.2.3.3.2

Rewrite the expression.

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

Step 5.2.3.4

Cancel the common factor of $2$ .

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Step 5.2.3.4.1

Cancel the common factor.

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

Step 5.2.3.4.2

Rewrite the expression.

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

Step 5.2.4

Combine the opposite terms in $\sin (\frac{x}{2}) + 1 - 1$ .

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Step 5.2.4.1

Subtract $1$ from $1$ .

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

Step 5.2.4.2

Add $\sin (\frac{x}{2})$ and $0$ .

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

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

$f (2 \arcsin (2 x - 1)) = \frac{1}{2} \cdot \sin (\frac{2 \arcsin (2 x - 1)}{2}) + \frac{1}{2}$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Cancel the common factor of $2$ .

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Step 5.3.3.1.1

Cancel the common factor.

$f (2 \arcsin (2 x - 1)) = \frac{1}{2} \cdot \sin (\frac{2 \arcsin (2 x - 1)}{2}) + \frac{1}{2}$

Step 5.3.3.1.2

Divide $\arcsin (2 x - 1)$ by $1$ .

$f (2 \arcsin (2 x - 1)) = \frac{1}{2} \cdot \sin (\arcsin (2 x - 1)) + \frac{1}{2}$

Step 5.3.3.2

The functions sine and arcsine are inverses.

$f (2 \arcsin (2 x - 1)) = \frac{1}{2} \cdot (2 x - 1) + \frac{1}{2}$

Step 5.3.3.3

Apply the distributive property.

$f (2 \arcsin (2 x - 1)) = \frac{1}{2} \cdot (2 x) + \frac{1}{2} \cdot - 1 + \frac{1}{2}$

Step 5.3.3.4

Cancel the common factor of $2$ .

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Step 5.3.3.4.1

Factor $2$ out of $2 x$ .

$f (2 \arcsin (2 x - 1)) = \frac{1}{2} \cdot (2 (x)) + \frac{1}{2} \cdot - 1 + \frac{1}{2}$

Step 5.3.3.4.2

Cancel the common factor.

$f (2 \arcsin (2 x - 1)) = \frac{1}{2} \cdot (2 x) + \frac{1}{2} \cdot - 1 + \frac{1}{2}$

Step 5.3.3.4.3

Rewrite the expression.

$f (2 \arcsin (2 x - 1)) = x + \frac{1}{2} \cdot - 1 + \frac{1}{2}$

Step 5.3.3.5

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

$f (2 \arcsin (2 x - 1)) = x + \frac{- 1}{2} + \frac{1}{2}$

Step 5.3.3.6

Move the negative in front of the fraction.

$f (2 \arcsin (2 x - 1)) = x - \frac{1}{2} + \frac{1}{2}$

Step 5.3.4

Combine the opposite terms in $x - \frac{1}{2} + \frac{1}{2}$ .

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Step 5.3.4.1

Add $- \frac{1}{2}$ and $\frac{1}{2}$ .

$f (2 \arcsin (2 x - 1)) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

$f (2 \arcsin (2 x - 1)) = x$

Step 5.4

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

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