Algebra Examples

$\arcsin (x)$

Interchange the variables.

$x = \arcsin (y)$

Solve for $y$ .

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

$\arcsin (y) = x$

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

$y = \sin (x)$

Remove parentheses.

$y = \sin (x)$

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

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

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

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

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

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Set up the composite result function.

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

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

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

The functions sine and arcsine are inverses.

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

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

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Set up the composite result function.

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

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

$f (\sin (x)) = \arcsin (\sin (x))$

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

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