Question

If R and C denote the set of real numbers and complex numbers, respectively. Then, the function \[f:C \to R\] defined by \[f(z) = |z|\] is
A. one – one
B. onto
C. bijective
D. neither one-one nor onto

Answer 1

Hint: Here the given question related to the relations and function. So we have to determine the what kind of function. The set of complex number is the domain of the function and the set of real numbers is the range of the function. So we check the definition of one – one function and onto function for the given function.

Formula Used:

Complete step by step Solution: The one – one function means for each element of the domain set will have unique image. Suppose, if the function have same image for more one element of domain set, then it is not considered as one – one function.
The onto function means every element in the range set will be the image.
If the function is both one – one and onto then it is called as bijective function.
On considering the given question.
The function \[f:C \to R\] defined by \[f(z) = |z|\]
Now we will consider two elements from the set of complex numbers i.e., 1 and -1. On applying the function
\[f(1) = |1| = 1\] and \[f( - 1) = | - 1| = 1\]
Here, we are getting the same image. Therefore the function is not one – one. On observing the given options only in the fourth option is satisfying our answer. So there is no need to check for the onto function.
Hence, the function is neither one – one nor onto.

Therefore, the option D is correct one.

Note: Usually student will think since the domain is the set of complex number then each and every element will be in the form of \[a + ib\]. The 1 and -1 can also be written in the form of \[a + ib\] i.e., \[1 = 1 + i.0\] and \[ - 1 = - 1 + i.0\]. Since it is multiple choice question we have to go through the options which can satisfying the solution obtained.