Find the domain of the function f and of its inverse function $\mathrm { f } ^ { - 1 }$ .
-$$f ( x ) = \cos ( x - 3 ) + 4$$
A) Domain of $f : ( \infty , \infty )$
Domain of $f ^ { - 1 } : [ - 5 , - 3 ]$

B) Domain of $\mathrm { f } : ( \infty , \infty )$
Domain of $\mathrm { f } ^ { - 1 } : ( \infty , \infty )$

C) Domain of $\mathrm { f } : ( \infty , \infty )$
Domain of $\mathrm { f } ^ { - 1 } : [ 3,5 ]$

D) Domain of $f : [ - 3,3 ]$

Domain of $\mathrm { f } ^ { - 1 } : ( \infty , \infty )$

Question

Find the domain of the function f and of its inverse function $\mathrm { f } ^ { - 1 }$ .
-$$f ( x ) = \cos ( x - 3 ) + 4$$
A) Domain of $f : ( \infty , \infty )$
Domain of $f ^ { - 1 } : [ - 5 , - 3 ]$

B) Domain of $\mathrm { f } : ( \infty , \infty )$
Domain of $\mathrm { f } ^ { - 1 } : ( \infty , \infty )$

C) Domain of $\mathrm { f } : ( \infty , \infty )$
Domain of $\mathrm { f } ^ { - 1 } : [ 3,5 ]$

D) Domain of $f : [ - 3,3 ]$

Domain of $\mathrm { f } ^ { - 1 } : ( \infty , \infty )$

Quizplus · Accepted Answer

The domain of $f(x) = \cos(x - 3) + 4$ is all real numbers because the cosine function can take any real number as its input, hence $(-\infty, \infty)$. The range of $f$, which becomes the domain of $f^{-1}$, is limited by the output of the cosine function, which varies between -1 and 1. After the transformation, the range of $f$ is between $3$ and $5$ ($4-1$ to $4+1$), so the domain of $f^{-1}$ is $[3, 5]$.

Find the Domain of the Function F and of Its f−1\mathrm { f } ^ { - 1 }f−1

Find the Domain of the Function F and of Its $\mathrm { f } ^ { - 1 }$