Find the Inverse -5cos(6x)

$- 5 \cos (6 x)$

Step 1

Interchange the variables.

$x = - 5 \cos (6 y)$

Step 2

Solve for $y$ .

Rewrite the equation as $- 5 \cos (6 y) = x$ .

$- 5 \cos (6 y) = x$

Divide each term in $- 5 \cos (6 y) = x$ by $- 5$ and simplify.

Divide each term in $- 5 \cos (6 y) = x$ by $- 5$ .

$\frac{- 5 \cos (6 y)}{- 5} = \frac{x}{- 5}$

Simplify the left side.

Cancel the common factor of $- 5$ .

Cancel the common factor.

$\frac{- 5 \cos (6 y)}{- 5} = \frac{x}{- 5}$

Divide $\cos (6 y)$ by $1$ .

$\cos (6 y) = \frac{x}{- 5}$

Simplify the right side.

Move the negative in front of the fraction.

$\cos (6 y) = - \frac{x}{5}$

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

$6 y = \arccos (- \frac{x}{5})$

Divide each term in $6 y = \arccos (- \frac{x}{5})$ by $6$ and simplify.

Divide each term in $6 y = \arccos (- \frac{x}{5})$ by $6$ .

$\frac{6 y}{6} = \frac{\arccos (- \frac{x}{5})}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 y}{6} = \frac{\arccos (- \frac{x}{5})}{6}$

Divide $y$ by $1$ .

$y = \frac{\arccos (- \frac{x}{5})}{6}$

Step 3

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

$f^{- 1} (x) = \frac{\arccos (- \frac{x}{5})}{6}$

Step 4

Verify if $f^{- 1} (x) = \frac{\arccos (- \frac{x}{5})}{6}$ is the inverse of $f (x) = - 5 \cos (6 x)$ .

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 $f^{- 1} (- 5 \cos (6 x))$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (- 5 \cos (6 x)) = \frac{\arccos (- \frac{- 5 \cos (6 x)}{5})}{6}$

Cancel the common factor of $- 5$ and $5$ .

Factor $5$ out of $- 5 \cos (6 x)$ .

$f^{- 1} (- 5 \cos (6 x)) = \frac{\arccos (- \frac{5 (- \cos (6 x))}{5})}{6}$

Cancel the common factors.

Factor $5$ out of $5$ .

$f^{- 1} (- 5 \cos (6 x)) = \frac{\arccos (- \frac{5 (- \cos (6 x))}{5 (1)})}{6}$

Cancel the common factor.

$f^{- 1} (- 5 \cos (6 x)) = \frac{\arccos (- \frac{5 (- \cos (6 x))}{5 \cdot 1})}{6}$

Rewrite the expression.

$f^{- 1} (- 5 \cos (6 x)) = \frac{\arccos (- \frac{- \cos (6 x)}{1})}{6}$

Divide $- \cos (6 x)$ by $1$ .

$f^{- 1} (- 5 \cos (6 x)) = \frac{\arccos (\cos (6 x))}{6}$

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

Set up the composite result function.

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

Evaluate $f (\frac{\arccos (- \frac{x}{5})}{6})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{\arccos (- \frac{x}{5})}{6}) = - 5 \cos (6 (\frac{\arccos (- \frac{x}{5})}{6}))$

Cancel the common factor of $6$ .

Cancel the common factor.

$f (\frac{\arccos (- \frac{x}{5})}{6}) = - 5 \cos (6 (\frac{\arccos (- \frac{x}{5})}{6}))$

Rewrite the expression.

$f (\frac{\arccos (- \frac{x}{5})}{6}) = - 5 \cos (\arccos (- \frac{x}{5}))$

The functions cosine and arccosine are inverses.

$f (\frac{\arccos (- \frac{x}{5})}{6}) = - 5 (- \frac{x}{5})$

Cancel the common factor of $5$ .

Move the leading negative in $- \frac{x}{5}$ into the numerator.

$f (\frac{\arccos (- \frac{x}{5})}{6}) = - 5 \frac{- x}{5}$

Factor $5$ out of $- 5$ .

$f (\frac{\arccos (- \frac{x}{5})}{6}) = 5 (- 1) (\frac{- x}{5})$

Cancel the common factor.

$f (\frac{\arccos (- \frac{x}{5})}{6}) = 5 \cdot (- 1 \frac{- x}{5})$

Rewrite the expression.

$f (\frac{\arccos (- \frac{x}{5})}{6}) = - 1 (- x)$

Multiply.

Multiply $- 1$ by $- 1$ .

$f (\frac{\arccos (- \frac{x}{5})}{6}) = 1 x$

Multiply $x$ by $1$ .

$f (\frac{\arccos (- \frac{x}{5})}{6}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{\arccos (- \frac{x}{5})}{6}$ is the inverse of $f (x) = - 5 \cos (6 x)$ .

$f^{- 1} (x) = \frac{\arccos (- \frac{x}{5})}{6}$

Find the Inverse -5cos(6x)

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