Precalculus Examples

$7 \sin (x) - 6$

Step 1

Interchange the variables.

$x = 7 \sin (y) - 6$

Step 2

Solve for $y$ .

Tap for more steps...

Step 2.1

Rewrite the equation as $7 \sin (y) - 6 = x$ .

$7 \sin (y) - 6 = x$

Step 2.2

Add $6$ to both sides of the equation.

$7 \sin (y) = x + 6$

Step 2.3

Divide each term in $7 \sin (y) = x + 6$ by $7$ and simplify.

Tap for more steps...

Step 2.3.1

Divide each term in $7 \sin (y) = x + 6$ by $7$ .

$\frac{7 \sin (y)}{7} = \frac{x}{7} + \frac{6}{7}$

Step 2.3.2

Simplify the left side.

Tap for more steps...

Step 2.3.2.1

Cancel the common factor of $7$ .

Tap for more steps...

Step 2.3.2.1.1

Cancel the common factor.

$\frac{7 \sin (y)}{7} = \frac{x}{7} + \frac{6}{7}$

Step 2.3.2.1.2

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

$\sin (y) = \frac{x}{7} + \frac{6}{7}$

Step 2.4

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

$y = \arcsin (\frac{x}{7} + \frac{6}{7})$

Step 3

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

$f^{- 1} (x) = \arcsin (\frac{x}{7} + \frac{6}{7})$

Step 4

Verify if $f^{- 1} (x) = \arcsin (\frac{x}{7} + \frac{6}{7})$ is the inverse of $f (x) = 7 \sin (x) - 6$ .

Tap for more steps...

Step 4.1

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

Step 4.2

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

Tap for more steps...

Step 4.2.1

Set up the composite result function.

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

Step 4.2.2

Evaluate $f^{- 1} (7 \sin (x) - 6)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (7 \sin (x) - 6) = \arcsin (\frac{7 \sin (x) - 6}{7} + \frac{6}{7})$

Step 4.2.3

Combine the numerators over the common denominator.

$f^{- 1} (7 \sin (x) - 6) = \arcsin (\frac{7 \sin (x) - 6 + 6}{7})$

Step 4.2.4

Combine the opposite terms in $7 \sin (x) - 6 + 6$ .

Tap for more steps...

Step 4.2.4.1

Add $- 6$ and $6$ .

$f^{- 1} (7 \sin (x) - 6) = \arcsin (\frac{7 \sin (x) + 0}{7})$

Step 4.2.4.2

Add $7 \sin (x)$ and $0$ .

$f^{- 1} (7 \sin (x) - 6) = \arcsin (\frac{7 \sin (x)}{7})$

Step 4.2.5

Cancel the common factor of $7$ .

Tap for more steps...

Step 4.2.5.1

Cancel the common factor.

$f^{- 1} (7 \sin (x) - 6) = \arcsin (\frac{7 \sin (x)}{7})$

Step 4.2.5.2

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

$f^{- 1} (7 \sin (x) - 6) = \arcsin (\sin (x))$

Step 4.3

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

Tap for more steps...

Step 4.3.1

Set up the composite result function.

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

Step 4.3.2

Evaluate $f (\arcsin (\frac{x}{7} + \frac{6}{7}))$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\arcsin (\frac{x}{7} + \frac{6}{7})) = 7 \sin (\arcsin (\frac{x}{7} + \frac{6}{7})) - 6$

Step 4.3.3

Simplify each term.

Tap for more steps...

Step 4.3.3.1

The functions sine and arcsine are inverses.

$f (\arcsin (\frac{x}{7} + \frac{6}{7})) = 7 (\frac{x}{7} + \frac{6}{7}) - 6$

Step 4.3.3.2

Apply the distributive property.

$f (\arcsin (\frac{x}{7} + \frac{6}{7})) = 7 (\frac{x}{7}) + 7 (\frac{6}{7}) - 6$

Step 4.3.3.3

Cancel the common factor of $7$ .

Tap for more steps...

Step 4.3.3.3.1

Cancel the common factor.

$f (\arcsin (\frac{x}{7} + \frac{6}{7})) = 7 (\frac{x}{7}) + 7 (\frac{6}{7}) - 6$

Step 4.3.3.3.2

Rewrite the expression.

$f (\arcsin (\frac{x}{7} + \frac{6}{7})) = x + 7 (\frac{6}{7}) - 6$

Step 4.3.3.4

Cancel the common factor of $7$ .

Tap for more steps...

Step 4.3.3.4.1

Cancel the common factor.

$f (\arcsin (\frac{x}{7} + \frac{6}{7})) = x + 7 (\frac{6}{7}) - 6$

Step 4.3.3.4.2

Rewrite the expression.

$f (\arcsin (\frac{x}{7} + \frac{6}{7})) = x + 6 - 6$

Step 4.3.4

Combine the opposite terms in $x + 6 - 6$ .

Tap for more steps...

Step 4.3.4.1

Subtract $6$ from $6$ .

$f (\arcsin (\frac{x}{7} + \frac{6}{7})) = x + 0$

Step 4.3.4.2

Add $x$ and $0$ .

$f (\arcsin (\frac{x}{7} + \frac{6}{7})) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \arcsin (\frac{x}{7} + \frac{6}{7})$ is the inverse of $f (x) = 7 \sin (x) - 6$ .

$f^{- 1} (x) = \arcsin (\frac{x}{7} + \frac{6}{7})$