Question

How do you find the domain and range of ln (inverse \[\sin x\] )?

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 sine 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:
ln (inverse \[\sin x\] )
Let,
 \[f\left( x \right) = \ln \left( {{{\sin }^{ - 1}}x} \right)\]
Now as \[\ln y\] is defined only if \[y > 0\] , therefore for \[\ln \left( {{{\sin }^{ - 1}}x} \right)\] to be defined we must have
 \[{\sin ^{ - 1}}x > 0\] ,
We know that \[{\sin ^{ - 1}}x\] is defined in the interval \[\left[ { - 1,1} \right] \] and \[{\sin ^{ - 1}}x > 0\] for \[x \in \left( {0,1} \right] \] .
Therefore, the domain of \[f\left( x \right)\] is \[x \in \left( {0,1} \right] \] .
And as \[{\sin ^{ - 1}}y\] and \[\ln y\] are increasing functions, the range of \[f\left( x \right)\] will be
 \[\left( { - \infty ,\ln \left( {\dfrac{\pi }{2}} \right)} \right] \] .
Hence,
Domain: \[\left( {0,1} \right] \]
Range: \[\left( { - \infty ,\ln \left( {\dfrac{\pi }{2}} \right)} \right] \]

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.