Finite Math Examples

$F (x) = \cos (x)$

Step 1

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

$y = \cos (x)$

Step 2

Interchange the variables.

$x = \cos (y)$

Step 3

Solve for $y$ .

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Step 3.1

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

$\cos (y) = x$

Step 3.2

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

$y = \arccos (x)$

Step 3.3

Remove parentheses.

$y = \arccos (x)$

Step 4

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

$F^{- 1} (x) = \arccos (x)$

Step 5

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

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

$F^{- 1} (\cos (x)) = \arccos (\cos (x))$

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

$F (\arccos (x)) = \cos (\arccos (x))$

Step 5.3.3

The functions cosine and arccosine are inverses.

$F (\arccos (x)) = x$

Step 5.4

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

$F^{- 1} (x) = \arccos (x)$