Find the inverse function $\mathrm { f } ^ { - 1 }$ of the function f.
-$$f ( x ) = - 4 \cos ( 10 x + 7 )$$
A) $f ^ { - 1 } ( x ) = - 4 \cos ^ { - 1 } ( 10 x + 7 )$
B) $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \frac { 1 } { 4 } \left[ \cos ^ { - 1 } \left( \frac { \mathrm { x } } { 10 } \right) - 7 \right]$
C) $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \frac { 1 } { 10 } \left[ \cos ^ { - 1 } \left( \frac { \mathrm { x } } { 4 } \right) - 7 \right]$
D) $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = - \frac { 1 } { 10 } \left[ \cos ^ { - 1 } \left( \frac { \mathrm { x } } { 4 } \right) + 7 \right]$

Question

Quizplus · Accepted Answer

To find the inverse function $f^{-1}(x)$, we swap $x$ and $y$ and solve for $y$. Starting with $y = -4\cos(10x + 7)$, swapping $x$ and $y$ gives $x = -4\cos(10y + 7)$. Solving for $y$, we isolate the cosine function: $\cos(10y + 7) = \frac{x}{-4}$. Then, applying the inverse cosine to both sides and solving for $y$ gives $y = \frac{1}{10}[\cos^{-1}(\frac{x}{-4}) - 7]$, which simplifies to the expression in choice C when considering the negative sign inside the inverse cosine does not affect the outcome due to the even nature of the cosine function.

Find the Inverse Function f−1\mathrm { f } ^ { - 1 }f−1

Find the Inverse Function $\mathrm { f } ^ { - 1 }$