Calculus Examples

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

Step 1

Write $f (x) = 2 \sin (x) - 1$ as an equation.

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $2 \sin (y) - 1 = x$ .

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

Step 3.2

Add $1$ to both sides of the equation.

$2 \sin (y) = x + 1$

Step 3.3

Divide each term in $2 \sin (y) = x + 1$ by $2$ and simplify.

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Step 3.3.1

Divide each term in $2 \sin (y) = x + 1$ by $2$ .

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

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $2$ .

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Step 3.3.2.1.1

Cancel the common factor.

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

Step 3.3.2.1.2

Divide $\sin (y)$ by $1$ .

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

Step 3.4

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

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

Step 4

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

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

Step 5

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

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

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

Step 5.2.3

Combine the numerators over the common denominator.

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

Step 5.2.4

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

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Step 5.2.4.1

Add $- 1$ and $1$ .

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

Step 5.2.4.2

Add $2 \sin (x)$ and $0$ .

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

Step 5.2.5

Cancel the common factor of $2$ .

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Step 5.2.5.1

Cancel the common factor.

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

Step 5.2.5.2

Divide $\sin (x)$ by $1$ .

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

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

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

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

The functions sine and arcsine are inverses.

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

Step 5.3.3.2

Apply the distributive property.

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

Step 5.3.3.3

Cancel the common factor of $2$ .

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Step 5.3.3.3.1

Cancel the common factor.

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

Step 5.3.3.3.2

Rewrite the expression.

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

Step 5.3.3.4

Cancel the common factor of $2$ .

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Step 5.3.3.4.1

Cancel the common factor.

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

Step 5.3.3.4.2

Rewrite the expression.

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

Step 5.3.4

Combine the opposite terms in $x + 1 - 1$ .

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Step 5.3.4.1

Subtract $1$ from $1$ .

$f (\arcsin (\frac{x}{2} + \frac{1}{2})) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

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

Step 5.4

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

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