Find the Inverse f(x)=5sin(x-7)

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Trigonometry

$f (x) = 5 \sin (x - 7)$

Replace $f (x)$ with $y$ .

$y = 5 \sin (x - 7)$

Interchange the variables.

$x = 5 \sin (y - 7)$

Solve for $y$ .

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Rewrite the equation as $5 \sin (y - 7) = x$ .

$5 \sin (y - 7) = x$

Divide each term by $5$ and simplify.

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Divide each term in $5 \sin (y - 7) = x$ by $5$ .

$\frac{5 \sin (y - 7)}{5} = \frac{x}{5}$

Cancel the common factor of $5$ .

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Cancel the common factor.

$\frac{5 \sin (y - 7)}{5} = \frac{x}{5}$

Divide $\sin (y - 7)$ by $1$ .

$\sin (y - 7) = \frac{x}{5}$

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

$y - 7 = \arcsin (\frac{x}{5})$

Add $7$ to both sides of the equation.

$y = \arcsin (\frac{x}{5}) + 7$

Solve for $y$ and replace with $f^{- 1} (x)$ .

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

$f^{- 1} (x) = \arcsin (\frac{x}{5}) + 7$

Set up the composite result function.

$g (f (x))$

Evaluate $g (f (x))$ by substituting in the value of $f$ into $g$ .

$g (5 \sin (x - 7)) = \arcsin (\frac{5 \sin (x - 7)}{5}) + 7$

Cancel the common factor of $5$ .

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Cancel the common factor.

$g (5 \sin (x - 7)) = \arcsin (\frac{5 \sin (x - 7)}{5}) + 7$

Divide $\sin (x - 7)$ by $1$ .

$g (5 \sin (x - 7)) = \arcsin (\sin (x - 7)) + 7$

Since $g (f (x)) = x$ , $f^{- 1} (x) = \arcsin (\frac{x}{5}) + 7$ is the inverse of $f (x) = 5 \sin (x - 7)$ .

$f^{- 1} (x) = \arcsin (\frac{x}{5}) + 7$

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