QuestionFind the inverse function $f^{-1}$ for $f(x)=\cos (x-4)-5$ .

Studdy Solution

STEP 1

Assumptions1. The function $f$ is given by $f(x)=\cos (x-4)-5$ . We are asked to find the inverse function $f^{-1}$ of the function $f$
3. We assume that the function $f$ is invertible, meaning there exists a function $f^{-1}$ such that $f^{-1}(f(x))=x$ for all $x$ in the domain of $f$ , and $f(f^{-1}(x))=x$ for all $x$ in the range of $f$ .

STEP 2

To find the inverse of a function, we first replace the function notation $f(x)$ with $y$ .
$y = \cos (x-4) -5$

STEP 3

Next, we swap $x$ and $y$ to reflect the fact that for the inverse function, the roles of $x$ and $y$ are reversed.
$x = \cos (y-) -5$

STEP 4

Now, we solve this equation for $y$ , which represents $f^{-1}(x)$ .
First, we isolate the cosine term by adding $$ to both sides.
$x + = \cos (y-4)$

STEP 5

Next, we take the inverse cosine of both sides to isolate $y-4$ .
$y-4 = \cos^{-1}(x+5)$

STEP 6

Finally, we add $4$ to both sides to solve for $y$ .
$y = \cos^{-1}(x+5) +4$

STEP 7

So, the inverse function $f^{-1}(x)$ is given by $f^{-1}(x) = \cos^{-1}(x+5) +4$ This is the solution to the problem.

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