Let's consider the line y = x, first, as that is the root of each transformation.
Now, let's understand the relationship between:
[tex]y = x^{2}[/tex]
[tex]y = \sqrt{x}[/tex]
We know that they are simply transformations of the line y = x.
But we know more than that. For a square root graph, we know that it can only take the positive x-integers on the real number line because, well, x has a restricted domain of [tex]x \geq 0[/tex].
For a quadratic, or degree 2 polynomial, we know that the domain will be unrestricted because any x-values will satisfy. So, we know that they will have different restrictions, and the two graphs will support that.
However, we can go one step ahead and compare the two equations to find the relationship. They are actually inverses of each other, when the quadratic restricts itself to become a function, because the parabola itself isn't a function.
Taking the positive side of the parabola, we get:
[tex]y = x^{2}, x \geq 0[/tex]
By interchanging the x and y-ordinates, or reflecting the graph about the line y = x, we get:
[tex]x = y^{2}, y \geq 0[/tex]
Solving for y:
[tex]y = \sqrt{x}, y \geq 0[/tex]
So, we know that the parabola and the square root of x graphs are inverses of each other. This means that these two are not the same.
Alternatively, if we compare them algebraically, we can conclude that they do not equal:
[tex]y = x^{2}[/tex]
[tex]y = \sqrt{x}[/tex]
[tex]x^{2} = \sqrt{x}[/tex]
[tex]x^{4} \neq x, x > 1[/tex]
Thus, we know that they only intersect at 0 and 1.
