Find the Inverse -4cos(7x)

$- 4 \cos (7 x)$

Step 1

Interchange the variables.

$x = - 4 \cos (7 y)$

Step 2

Solve for $y$ .

Rewrite the equation as $- 4 \cos (7 y) = x$ .

$- 4 \cos (7 y) = x$

Divide each term in $- 4 \cos (7 y) = x$ by $- 4$ and simplify.

Divide each term in $- 4 \cos (7 y) = x$ by $- 4$ .

$\frac{- 4 \cos (7 y)}{- 4} = \frac{x}{- 4}$

Simplify the left side.

Cancel the common factor of $- 4$ .

Cancel the common factor.

$\frac{- 4 \cos (7 y)}{- 4} = \frac{x}{- 4}$

Divide $\cos (7 y)$ by $1$ .

$\cos (7 y) = \frac{x}{- 4}$

Simplify the right side.

Move the negative in front of the fraction.

$\cos (7 y) = - \frac{x}{4}$

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

$7 y = \arccos (- \frac{x}{4})$

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

Divide each term in $7 y = \arccos (- \frac{x}{4})$ by $7$ .

$\frac{7 y}{7} = \frac{\arccos (- \frac{x}{4})}{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 y}{7} = \frac{\arccos (- \frac{x}{4})}{7}$

Divide $y$ by $1$ .

$y = \frac{\arccos (- \frac{x}{4})}{7}$

Step 3

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

$f^{- 1} (x) = \frac{\arccos (- \frac{x}{4})}{7}$

Step 4

Verify if $f^{- 1} (x) = \frac{\arccos (- \frac{x}{4})}{7}$ is the inverse of $f (x) = - 4 \cos (7 x)$ .

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

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

Set up the composite result function.

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

Evaluate $f^{- 1} (- 4 \cos (7 x))$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (- 4 \cos (7 x)) = \frac{\arccos (- \frac{- 4 \cos (7 x)}{4})}{7}$

Cancel the common factor of $- 4$ and $4$ .

Factor $4$ out of $- 4 \cos (7 x)$ .

$f^{- 1} (- 4 \cos (7 x)) = \frac{\arccos (- \frac{4 (- \cos (7 x))}{4})}{7}$

Cancel the common factors.

Factor $4$ out of $4$ .

$f^{- 1} (- 4 \cos (7 x)) = \frac{\arccos (- \frac{4 (- \cos (7 x))}{4 (1)})}{7}$

Cancel the common factor.

$f^{- 1} (- 4 \cos (7 x)) = \frac{\arccos (- \frac{4 (- \cos (7 x))}{4 \cdot 1})}{7}$

Rewrite the expression.

$f^{- 1} (- 4 \cos (7 x)) = \frac{\arccos (- \frac{- \cos (7 x)}{1})}{7}$

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

$f^{- 1} (- 4 \cos (7 x)) = \frac{\arccos (\cos (7 x))}{7}$

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

Set up the composite result function.

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

Evaluate $f (\frac{\arccos (- \frac{x}{4})}{7})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{\arccos (- \frac{x}{4})}{7}) = - 4 \cos (7 (\frac{\arccos (- \frac{x}{4})}{7}))$

Cancel the common factor of $7$ .

Cancel the common factor.

$f (\frac{\arccos (- \frac{x}{4})}{7}) = - 4 \cos (7 (\frac{\arccos (- \frac{x}{4})}{7}))$

Rewrite the expression.

$f (\frac{\arccos (- \frac{x}{4})}{7}) = - 4 \cos (\arccos (- \frac{x}{4}))$

The functions cosine and arccosine are inverses.

$f (\frac{\arccos (- \frac{x}{4})}{7}) = - 4 (- \frac{x}{4})$

Cancel the common factor of $4$ .

Move the leading negative in $- \frac{x}{4}$ into the numerator.

$f (\frac{\arccos (- \frac{x}{4})}{7}) = - 4 \frac{- x}{4}$

Factor $4$ out of $- 4$ .

$f (\frac{\arccos (- \frac{x}{4})}{7}) = 4 (- 1) (\frac{- x}{4})$

Cancel the common factor.

$f (\frac{\arccos (- \frac{x}{4})}{7}) = 4 \cdot (- 1 \frac{- x}{4})$

Rewrite the expression.

$f (\frac{\arccos (- \frac{x}{4})}{7}) = - 1 (- x)$

Multiply.

Multiply $- 1$ by $- 1$ .

$f (\frac{\arccos (- \frac{x}{4})}{7}) = 1 x$

Multiply $x$ by $1$ .

$f (\frac{\arccos (- \frac{x}{4})}{7}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{\arccos (- \frac{x}{4})}{7}$ is the inverse of $f (x) = - 4 \cos (7 x)$ .

$f^{- 1} (x) = \frac{\arccos (- \frac{x}{4})}{7}$

Find the Inverse -4cos(7x)

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