Precalculus Examples

$y = \arccos (\frac{x}{2})$

Step 1

Interchange the variables.

$x = \arccos (\frac{y}{2})$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\arccos (\frac{y}{2}) = x$ .

$\arccos (\frac{y}{2}) = x$

Step 2.2

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

$\frac{y}{2} = \cos (x)$

Step 2.3

Multiply both sides of the equation by $2$ .

$2 \frac{y}{2} = 2 \cos (x)$

Step 2.4

Simplify the left side.

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Step 2.4.1

Cancel the common factor of $2$ .

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Step 2.4.1.1

Cancel the common factor.

$2 \frac{y}{2} = 2 \cos (x)$

Step 2.4.1.2

Rewrite the expression.

$y = 2 \cos (x)$

Step 3

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

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

Step 4

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

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Step 4.1

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

Step 4.2

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

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Step 4.2.1

Set up the composite result function.

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

Step 4.2.2

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

$f^{- 1} (\arccos (\frac{x}{2})) = 2 \cos (\arccos (\frac{x}{2}))$

Step 4.2.3

The functions cosine and arccosine are inverses.

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

Step 4.2.4

Cancel the common factor of $2$ .

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Step 4.2.4.1

Cancel the common factor.

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

Step 4.2.4.2

Rewrite the expression.

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

Step 4.3

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

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Step 4.3.1

Set up the composite result function.

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

Step 4.3.2

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

$f (2 \cos (x)) = \arccos (\frac{2 \cos (x)}{2})$

Step 4.3.3

Cancel the common factor of $2$ .

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Step 4.3.3.1

Cancel the common factor.

$f (2 \cos (x)) = \arccos (\frac{2 \cos (x)}{2})$

Step 4.3.3.2

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

$f (2 \cos (x)) = \arccos (\cos (x))$

Step 4.4

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

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