Question: If R and C denote the set of real numbers and complex numbers, respectively. Then, the function \[f:...

Question

If R and C denote the set of real numbers and complex numbers, respectively. Then, the function $f:C\to R$ defined by $f(z)=\left| z \right|$ is
1. One-one
2. Onto
3. Bijective
4. Neither one-one nor onto

Answer 1

Solution

In this particular problem the function is given that $f(z)=\left| z \right|$ and that we have to check whether a function is defined by onto or one-one or bijective or neither one to one function. So we can check this function by considering complex numbers and check the condition $f(x)=f(-x)$ so, in this way we have to solve this by using this approach for such a problem.

Complete step-by-step solution:
Before we go to the problem first of all we need to have a basic understanding of one-one, onto, bijective.
One-one function is also known as an injective function. One element contains only one elements that is one-one correspondences
Onto function is nothing but a many-one correspondence. It is also known as surjective function.
Bijective function is a function having both surjective as well as objective function.
Now, we go the problem in problem function is given $f(z)=\left| z \right|$
As we know that $\left| z \right|=\sqrt{{{x}^{2}}+{{y}^{2}}}$
So, first of all we have to check this condition $f(z)=f(-z)$ if this condition satisfies then it has neither one-one nor onto function.
To defined the function we need to consider two complex numbers $3+4i$ and $-3-4i$
$f(3+4i)=\left| 3+4i \right|$
By applying formula we get:
$f(3+4i)=\sqrt{{{3}^{2}}+{{4}^{2}}}$
Simplify further we get:
$f(3+4i)=5-----(1)$
Similarly for $f(-3-4i)=\left| -3-4i \right|$
By applying formula we get:
$f(4+3i)=\sqrt{{{-3}^{2}}+{{-4}^{2}}}$
Simplify further we get:
$f(4+3i)=5--------(2)$
From equation (1) and (2) we get:
Clearly $f(z)$ is not injective and also $f(z)$ is not surjective either as the negative real numbers are not the images of any z.
And also $f(z)=f(-z)$ this satisfies the conditions.
So, the correct option is “option 4”.

Note: In this type of problems always remember the condition to define whether it is bijective or not. For that we need to consider some function to check the function defined as we have seen in the above solution. Don’t confuse the function of onto one-one or bijective and don’t make silly mistakes while simplifying or checking the conditions. So, the above solution is preferred for such types of problems.