Algebra Examples

$f (x) = 4 \sin (2 x + 5) - 7$

Step 1

Write $f (x) = 4 \sin (2 x + 5) - 7$ as an equation.

$y = 4 \sin (2 x + 5) - 7$

Step 2

Interchange the variables.

$x = 4 \sin (2 y + 5) - 7$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $4 \sin (2 y + 5) - 7 = x$ .

$4 \sin (2 y + 5) - 7 = x$

Step 3.2

Add $7$ to both sides of the equation.

$4 \sin (2 y + 5) = x + 7$

Step 3.3

Divide each term in $4 \sin (2 y + 5) = x + 7$ by $4$ and simplify.

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Step 3.3.1

Divide each term in $4 \sin (2 y + 5) = x + 7$ by $4$ .

$\frac{4 \sin (2 y + 5)}{4} = \frac{x}{4} + \frac{7}{4}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $4$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{4 \sin (2 y + 5)}{4} = \frac{x}{4} + \frac{7}{4}$

Step 3.3.2.1.2

Divide $\sin (2 y + 5)$ by $1$ .

$\sin (2 y + 5) = \frac{x}{4} + \frac{7}{4}$

Step 3.4

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

$2 y + 5 = \arcsin (\frac{x}{4} + \frac{7}{4})$

Step 3.5

Subtract $5$ from both sides of the equation.

$2 y = \arcsin (\frac{x}{4} + \frac{7}{4}) - 5$

Step 3.6

Divide each term in $2 y = \arcsin (\frac{x}{4} + \frac{7}{4}) - 5$ by $2$ and simplify.

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Step 3.6.1

Divide each term in $2 y = \arcsin (\frac{x}{4} + \frac{7}{4}) - 5$ by $2$ .

$\frac{2 y}{2} = \frac{\arcsin (\frac{x}{4} + \frac{7}{4})}{2} + \frac{- 5}{2}$

Step 3.6.2

Simplify the left side.

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Step 3.6.2.1

Cancel the common factor of $2$ .

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Step 3.6.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{\arcsin (\frac{x}{4} + \frac{7}{4})}{2} + \frac{- 5}{2}$

Step 3.6.2.1.2

Divide $y$ by $1$ .

$y = \frac{\arcsin (\frac{x}{4} + \frac{7}{4})}{2} + \frac{- 5}{2}$

Step 3.6.3

Simplify the right side.

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Step 3.6.3.1

Move the negative in front of the fraction.

$y = \frac{\arcsin (\frac{x}{4} + \frac{7}{4})}{2} - \frac{5}{2}$

Step 4

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

$f^{- 1} (x) = \frac{\arcsin (\frac{x}{4} + \frac{7}{4})}{2} - \frac{5}{2}$

Step 5

Verify if $f^{- 1} (x) = \frac{\arcsin (\frac{x}{4} + \frac{7}{4})}{2} - \frac{5}{2}$ is the inverse of $f (x) = 4 \sin (2 x + 5) - 7$ .

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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 $f^{- 1} (4 \sin (2 x + 5) - 7)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (4 \sin (2 x + 5) - 7) = \frac{\arcsin (\frac{4 \sin (2 x + 5) - 7}{4} + \frac{7}{4})}{2} - \frac{5}{2}$

Step 5.2.3

Simplify the numerator.

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Step 5.2.3.1

Combine the numerators over the common denominator.

$f^{- 1} (4 \sin (2 x + 5) - 7) = \frac{\arcsin (\frac{4 \sin (2 x + 5) - 7 + 7}{4})}{2} - \frac{5}{2}$

Step 5.2.3.2

Combine the opposite terms in $4 \sin (2 x + 5) - 7 + 7$ .

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Step 5.2.3.2.1

Add $- 7$ and $7$ .

$f^{- 1} (4 \sin (2 x + 5) - 7) = \frac{\arcsin (\frac{4 \sin (2 x + 5) + 0}{4})}{2} - \frac{5}{2}$

Step 5.2.3.2.2

Add $4 \sin (2 x + 5)$ and $0$ .

$f^{- 1} (4 \sin (2 x + 5) - 7) = \frac{\arcsin (\frac{4 \sin (2 x + 5)}{4})}{2} - \frac{5}{2}$

Step 5.2.3.3

Cancel the common factor of $4$ .

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Step 5.2.3.3.1

Cancel the common factor.

$f^{- 1} (4 \sin (2 x + 5) - 7) = \frac{\arcsin (\frac{4 \sin (2 x + 5)}{4})}{2} - \frac{5}{2}$

Step 5.2.3.3.2

Divide $\sin (2 x + 5)$ by $1$ .

$f^{- 1} (4 \sin (2 x + 5) - 7) = \frac{\arcsin (\sin (2 x + 5))}{2} - \frac{5}{2}$

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 $f (\frac{\arcsin (\frac{x}{4} + \frac{7}{4})}{2} - \frac{5}{2})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{\arcsin (\frac{x}{4} + \frac{7}{4})}{2} - \frac{5}{2}) = 4 \sin (2 (\frac{\arcsin (\frac{x}{4} + \frac{7}{4})}{2} - \frac{5}{2}) + 5) - 7$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Simplify each term.

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Step 5.3.3.1.1

Combine the numerators over the common denominator.

$f (\frac{\arcsin (\frac{x}{4} + \frac{7}{4})}{2} - \frac{5}{2}) = 4 \sin (2 (\frac{\arcsin (\frac{x + 7}{4})}{2} - \frac{5}{2}) + 5) - 7$

Step 5.3.3.1.2

Combine the numerators over the common denominator.

$f (\frac{\arcsin (\frac{x}{4} + \frac{7}{4})}{2} - \frac{5}{2}) = 4 \sin (2 (\frac{\arcsin (\frac{x + 7}{4}) - 5}{2}) + 5) - 7$

Step 5.3.3.1.3

Cancel the common factor of $2$ .

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Step 5.3.3.1.3.1

Cancel the common factor.

$f (\frac{\arcsin (\frac{x}{4} + \frac{7}{4})}{2} - \frac{5}{2}) = 4 \sin (2 (\frac{\arcsin (\frac{x + 7}{4}) - 5}{2}) + 5) - 7$

Step 5.3.3.1.3.2

Rewrite the expression.

$f (\frac{\arcsin (\frac{x}{4} + \frac{7}{4})}{2} - \frac{5}{2}) = 4 \sin (\arcsin (\frac{x + 7}{4}) - 5 + 5) - 7$

Step 5.3.3.2

Combine the opposite terms in $\arcsin (\frac{x + 7}{4}) - 5 + 5$ .

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Step 5.3.3.2.1

Add $- 5$ and $5$ .

$f (\frac{\arcsin (\frac{x}{4} + \frac{7}{4})}{2} - \frac{5}{2}) = 4 \sin (\arcsin (\frac{x + 7}{4}) + 0) - 7$

Step 5.3.3.2.2

Add $\arcsin (\frac{x + 7}{4})$ and $0$ .

$f (\frac{\arcsin (\frac{x}{4} + \frac{7}{4})}{2} - \frac{5}{2}) = 4 \sin (\arcsin (\frac{x + 7}{4})) - 7$

Step 5.3.3.3

The functions sine and arcsine are inverses.

$f (\frac{\arcsin (\frac{x}{4} + \frac{7}{4})}{2} - \frac{5}{2}) = 4 (\frac{x + 7}{4}) - 7$

Step 5.3.3.4

Cancel the common factor of $4$ .

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Step 5.3.3.4.1

Cancel the common factor.

$f (\frac{\arcsin (\frac{x}{4} + \frac{7}{4})}{2} - \frac{5}{2}) = 4 (\frac{x + 7}{4}) - 7$

Step 5.3.3.4.2

Rewrite the expression.

$f (\frac{\arcsin (\frac{x}{4} + \frac{7}{4})}{2} - \frac{5}{2}) = x + 7 - 7$

Step 5.3.4

Combine the opposite terms in $x + 7 - 7$ .

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Step 5.3.4.1

Subtract $7$ from $7$ .

$f (\frac{\arcsin (\frac{x}{4} + \frac{7}{4})}{2} - \frac{5}{2}) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

$f (\frac{\arcsin (\frac{x}{4} + \frac{7}{4})}{2} - \frac{5}{2}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{\arcsin (\frac{x}{4} + \frac{7}{4})}{2} - \frac{5}{2}$ is the inverse of $f (x) = 4 \sin (2 x + 5) - 7$ .

$f^{- 1} (x) = \frac{\arcsin (\frac{x}{4} + \frac{7}{4})}{2} - \frac{5}{2}$