If the function: , is defined by , then which of the following statements is TRUE?
Function is both one-one and onto
Explanation for the correct option
Step 1: Identify the nature of the given function
Given function,
Identifying if the function is odd, by applying the condition
So, for the given function. This also demonstrates that the function is continuous.
Thus, is an odd, non-periodic continuous function.
Step 2: Evaluate the domain and range of the function
The function can be written as
So,
and,
Thus, the range of and is an onto function.
Differentiate the given function, , we get,
, where the result will always be positive or zero.
So,
Thus, is a one-one function.
Therefore, option (C) i.e. Function is both one-one and onto, is the correct answer.