Question

For the following, can $y$ be a function of $x,(xϵR,yϵR)$?

$x^6=y^3$

If it is a function then write 1 otherwise write 0.

Answer 1

$Graphically:$

$\text{If we draw a line parallel to y-axis and if it cuts the curve at more than one point, then it is not a function}$

$\text{If we draw a line parallel to y-axis and if it cuts the curve at only one point, then it is a function}$

$Bydefinition:$ A function is an equation for which any x that can be plugged into the equation will yield exactly one y out of the equation.

$Otherdefinition$ 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.

If we draw a line parallel to y-axis, it cuts the curve at single points.

$\therefore$ Given graph represent a function.

Also, Given for $x,y\in R$
$x^6=y^3$
$\Rightarrow x^2=y$
$\Rightarrow y=x^2$

So, for every $x\in R$, there is a unique $y\in R$
Hence, $y$ is a function of $x$