Question

How do you find the domain and range of inverse \[\cos \left( {{e^x}} \right)\]?

Answer 1

Hint: The domain of a function is the set of all acceptable input values (x-values). The range of a function is the set of all output values (y-values) and to find the domain and range of a given trigonometric function we must know the range and domain of the cos function, hence by applying the range and domain of trigonometric functions we can solve the given function.

Complete step-by-step answer:
 Let us write the given function:
Let,
\[f\left( x \right) = \cos \left( {{e^x}} \right)\]
or \[y = \cos \left( {{e^x}} \right)\]
We need to make x the subject of \[y = \cos \left( {{e^x}} \right)\]
\[{\cos ^{ - 1}}\left( y \right) = {\cos ^{ - 1}}\left( {\cos \left( {{e^x}} \right)} \right)\]
\[ \Rightarrow \]\[{\cos ^{ - 1}}\left( y \right) = {e^x}\]
Taking natural logs on both sides as:
\[\ln \left( {{{\cos }^{ - 1}}\left( y \right)} \right) = x\ln \left( e \right)\]
Now dividing by \[\ln e\]
\[\dfrac{{\ln \left( {{{\cos }^{ - 1}}\left( y \right)} \right)}}{{\ln e}} = \dfrac{{x\ln \left( e \right)}}{{\ln e}}\]
\[ \Rightarrow \]\[\dfrac{{\ln \left( {{{\cos }^{ - 1}}\left( y \right)} \right)}}{{\ln e}} = x\]
We know that \[\ln e\] = 1 (logarithm of the base is always 1)
Hence, we get:
\[x = \ln \left( {{{\cos }^{ - 1}}\left( y \right)} \right)\]
\[ \Rightarrow \]\[{f^{ - 1}}\left( x \right) = \ln \left( {{{\cos }^{ - 1}}\left( x \right)} \right)\]
\[y = \ln \left( {{{\cos }^{ - 1}}\left( x \right)} \right)\]
We know that,
\[{\cos ^{ - 1}}\left( x \right) \geqslant 0\]
\[ \Rightarrow \]\[ - 1 \leqslant x \leqslant 1\]
Hence,
Domain is: \[\left\{ {x \in R: - 1 \leqslant x \leqslant 1} \right\}\]and
Range:
Maximum value of \[{\cos ^{ - 1}}\left( x \right) = \pi \], when x = -1
\[ \Rightarrow \]\[\ln \pi \]
Minimum value of \[\ln {\cos ^{ - 1}}\left( x \right) \to - \infty \], when \[x \to 1\]
\[ \Rightarrow \]\[\left\{ {x \in R: - \infty \leqslant y \leqslant \ln \pi } \right\}\]

Note: The domain refers to the set of possible input values, the domain of a graph consists of all the input values shown on the x-axis. They may also have been called the input and output of the function. The domain of a function f(x) is the set of all values for which the function is defined, and the range of the function is the set of all values that f takes.