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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Name

Name	one billion four hundred ninety-seven million nine hundred forty-three thousand four hundred fifty-six

Interesting facts

1497943456 has 64 divisors, whose sum is 11375008362
The reverse of 1497943456 is 6543497941
Previous prime number is 2

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 52
Digital Root 7

Name

Name	five hundred sixty-two million six hundred fifty-two thousand five hundred ninety-six

Interesting facts

562652596 has 32 divisors, whose sum is 1384799328
The reverse of 562652596 is 695256265
Previous prime number is 373

Basic properties

Is Prime? no
Number parity even
Number length 9
Sum of Digits 46
Digital Root 1

Name

Name	six hundred ninety-four million four hundred three thousand four hundred eleven

Interesting facts

694403411 has 4 divisors, whose sum is 747819072
The reverse of 694403411 is 114304496
Previous prime number is 13

Basic properties

Is Prime? no
Number parity odd
Number length 9
Sum of Digits 32
Digital Root 5

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