Precalculus Examples

$y = \cos (2 x)$

Step 1

Interchange the variables.

$x = \cos (2 y)$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\cos (2 y) = x$ .

$\cos (2 y) = x$

Step 2.2

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

$2 y = \arccos (x)$

Step 2.3

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

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Step 2.3.1

Divide each term in $2 y = \arccos (x)$ by $2$ .

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

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $2$ .

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Step 2.3.2.1.1

Cancel the common factor.

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

Step 2.3.2.1.2

Divide $y$ by $1$ .

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

Step 3

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

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

Step 4

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

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

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

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

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

Step 4.3.3

Cancel the common factor of $2$ .

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Step 4.3.3.1

Cancel the common factor.

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

Step 4.3.3.2

Rewrite the expression.

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

Step 4.3.4

The functions cosine and arccosine are inverses.

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

Step 4.4

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

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