Question

For \[{{h}_{1}}\left( x \right)\] and \[{{h}_{2}}\left( x \right)\]are identical functions, then which of the following is not true?
a). Domain of \[{{h}_{1}}\left( x \right)\] and \[{{h}_{2}}\left( x \right)\]; $x\in [2n\pi ,(2n+1)\pi ]$ $n\in Z$
b). Range of \[{{h}_{1}}\left( x \right)\] and \[{{h}_{2}}\left( x \right)\] is $[0,1]$
c). Period of \[{{h}_{1}}\left( x \right)\] and \[{{h}_{2}}\left( x \right)\] is $\pi $
d). None of these

Answer 1

Hint: For the type of question we should know what is an Identical function. Two functions are called an Identical function if their domain and co-domain are the same. As we had to know which option does not hold true for an identical function. So we just need to go through the option and check which option does not fit for an identical function. Whichever stands out will be the answer, and if not then none of these will be the answer.

Complete step-by-step solution:
Moving further with the question as mentioned about two function that are \[{{h}_{1}}\left( x \right)\] and \[{{h}_{2}}\left( x \right)\]
Cross checking with the options.
Let us first go with option ‘a’ i.e. Domain of \[{{h}_{1}}\left( x \right)\] and \[{{h}_{2}}\left( x \right)\]; $x\in [2n\pi ,(2n+1)\pi ]$ $n\in Z$Since, the options says that both functions have same domain i.e. $x\in [2n\pi ,(2n+1)\pi ]$, which is a condition of an identical function. So this option is correct. So this is not the answer.
Moving forward with the option ‘b’ i.e. Range of \[{{h}_{1}}\left( x \right)\] and \[{{h}_{2}}\left( x \right)\]is $[0,1]$; Since, the options says that both functions have same co-domain i.e. $x\in [2n\pi ,(2n+1)\pi ]$, which is a condition of an identical function. So this option is correct. So this is also not the answer.
Moving forward with option ‘c’ i.e. Period of \[{{h}_{1}}\left( x \right)\] and \[{{h}_{2}}\left( x \right)\]is $\pi $; As options says both functions have the same period. Which is not a ‘must’ condition of an identical function as we had with domain and codomain. Means it may have the same period or not.
So, we can say that option c is wrong, because we cannot directly point out that the two identical function that are \[{{h}_{1}}\left( x \right)\] and \[{{h}_{2}}\left( x \right)\]will have same period; as we said in option ‘a’ and option ‘b’. So option c is wrong. And we are asked which one is not true for an identical function. So option ‘c’ is not true for an identical function.

Note: Keep in mind that co-domain and range are almost similar things, and here they both are the same. So if instead of co-domain, the term range or vice versa is used then assume it to be the same.