Question

How do you decide whether the relation ${y^2} = 4x$ defines a function?

Answer 1

We will first give the definition of function and then we will look out for the given function whether it follows all the mentioned conditions or not.

Complete step-by-step solution:
We are given that we need to decide whether the relation ${y^2} = 4x$ defines a function or not.
Let us first of all see the definition of function:-
Function:- A function is a binary relation between two sets that associates every element of the first set to exactly one element of the second set.
Here, we see for any non - negative value of y, we get two possible values of y, one positive and one negative.
For example:- take x = 4, then y might be 4 or -4.
Therefore, it is not a function. However, if we bound the range of the function to positive real numbers and zero, then multiple images would not be a problem.
Now, if you put in any negative value of x, we will get an imaginary value of y which does not exist in real numbers, therefore, we will also have to restrict our domain to positive real numbers including zero as well.
So, if we define the above relation from {\mathbb{R}^ + } \cup \left\\{ 0 \right\\} \to {\mathbb{R}^ + } \cup \left\\{ 0 \right\\}, then we can say that the given relation is a function.

Note: The students must know that the definition of function might seem easy but be a bit difficult to understand. We have to make sure of the fact that every point in the domain is covered with some element in the range and also that every point in the domain has exactly one image. However, it is possible that an element has more than one pre – image which means more than one element has the same image or there might also be some extra elements in the co – domain set as well.