Calculus Examples

$f (x) = \cos (9 x)$

Step 1

Write $f (x) = \cos (9 x)$ as an equation.

$y = \cos (9 x)$

Step 2

Interchange the variables.

$x = \cos (9 y)$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\cos (9 y) = x$ .

$\cos (9 y) = x$

Step 3.2

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

$9 y = \arccos (x)$

Step 3.3

Divide each term in $9 y = \arccos (x)$ by $9$ and simplify.

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Step 3.3.1

Divide each term in $9 y = \arccos (x)$ by $9$ .

$\frac{9 y}{9} = \frac{\arccos (x)}{9}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $9$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{9 y}{9} = \frac{\arccos (x)}{9}$

Step 3.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{\arccos (x)}{9}$

Step 4

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

$f^{- 1} (x) = \frac{\arccos (x)}{9}$

Step 5

Verify if $f^{- 1} (x) = \frac{\arccos (x)}{9}$ is the inverse of $f (x) = \cos (9 x)$ .

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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 $f^{- 1} (\cos (9 x))$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\cos (9 x)) = \frac{\arccos (\cos (9 x))}{9}$

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 $f (\frac{\arccos (x)}{9})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{\arccos (x)}{9}) = \cos (9 (\frac{\arccos (x)}{9}))$

Step 5.3.3

Cancel the common factor of $9$ .

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Step 5.3.3.1

Cancel the common factor.

$f (\frac{\arccos (x)}{9}) = \cos (9 (\frac{\arccos (x)}{9}))$

Step 5.3.3.2

Rewrite the expression.

$f (\frac{\arccos (x)}{9}) = \cos (\arccos (x))$

Step 5.3.4

The functions cosine and arccosine are inverses.

$f (\frac{\arccos (x)}{9}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{\arccos (x)}{9}$ is the inverse of $f (x) = \cos (9 x)$ .

$f^{- 1} (x) = \frac{\arccos (x)}{9}$