Question

Why does a radical function with an even index only appear on one sode of the x-axis while a radical woth an odd index appears on both sides?

Answer 1

Consider the equation y = x^2. No matter what x happens to be, the result y will never be negative even if x is negative. Example: x = -3 leads to y = x^2 = (-3)^2 = 9 which is positive.

Since y is never negative, this means the inverse x = sqrt(y) has the right hand side never be negative. The entire curve of sqrt(x) is above the x axis except for the x intercept of course. Put another way, we cannot plug in a negative input into the square root function for this reason. This similar idea applies to any even index such as fourth roots or sixth roots.

Meanwhile, odd roots such as a cube root has its range extend from negative infinity to positive infinity. Why? Because y = x^3 can have a negative output. Going back to x = -3 we get y = x^3 = (-3)^3 = -27. So we can plug a negative value into the cube root to get some negative output. We can get any output we want, negative or positive. So the range of any radical with an odd index is effectively the set of all real numbers. Visually this produces graphs that have parts on both sides of the x axis.

Answer 2

Let us Consider the function $y = x^2.$

For any value of x ,the value of y will never be negative.

For example

If x = -1

then y = 1 which is positive.

Since y is never negative, this means that the graph of the function will lies in the positive y axis that is index only appear on one side of the x-axis.

This is true for any even index.

Now consider the cubic function $y=x^{3}$

As the odd function can have a negative y value

For example

If x = -3 we get y = -1.

So the range of any radical with an odd index is the set of all real numbers. The cubic function will have a graph that have parts on both sides of the x axis.

Therefore, a radical with an odd index appears on both sides.

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