Question

For the following, $y$ can be a function of $x$, $\left( {x \in R,y \in R} \right)?$
${y^2} = 4ax$

Answer 1

Hint:A function is a relation from a set of inputs to a set of possible outputs where each input is related to exactly one output. It is denoted as $f:{\text{X}} \to {\text{Y}}$, a function from X to Y.
It means that if the object x is in the set of inputs (called the domain) then the function f will relate to exactly one object f(x).

Complete step-by-step answer:
A relation between two sets is a collection of ordered pairs containing one object from each set.
A function is a relation for which each value from the set the first components of the ordered pairs is associated with exactly one value from the set of second components of the ordered pair.
Given, $x \in R,y \in R$ and a relation in between x and y as: \[{y^2} = 4ax \Rightarrow y = 2\sqrt {ax} .......\left( i \right)\]
Equation (i) represents for every value of x, there exists one value of y.
Thus, by definition of function, each set of values of inputs has a set of possible outputs, i.e., If $x = 0$ , then $y = 2\sqrt {a.0} = 0$
If $x = 1$, then $y = 2\sqrt {a.1} = 2\sqrt a $
If $x = a$, then $y = 2\sqrt {a.a} = 2a$…… etc.
Hence, ${y^2} = 4ax$ is a function of x.

Note:To determine whether a relation is a function, by using the vertical line test on the graph of a relation (in between two variables).If a vertical line crosses the relation on the graph only once in all locations, the relation is a function. And if a vertical line crosses the relation more than once, the relation is not a function.
Also, please note that a given function is an equation of parabola whose focus is (0, 0) and the graph of it is symmetrical about the x – axis so it is called the axis of symmetry for such equations of parabola. Graphs of parabolas are always U-shaped curves, which may be symmetrical about X- axis or negative X – axis or Y – axis or negative Y – axis.