Functions of Squares and Square Roots?

y = x^2, y = x^3 and y = x^4 are all functions. Remember, there are two ways to tell a function: 1) If the graph passes the vertical line test (i.e. a vertical line drawn anywhere will never pass through the graph of a function more than once) or 2) if the function passes the ambiguity test, i.e. the is no x-value which gives you more than one result.

For the others, it depends.

Typically, "the square root of x" means only the positive square root. If that's the sense, then yes - the square root of x is a function. If, on the other hand, you consider the square root of, say, 4 to be either positive or negative 2, then it's not a function, because one value of x (4) gives you more than one value for y (+/- 2).

The cube root would be a function in either sense because you never get more than one answer.

The fourth root might or might not, for the same reason as the square root.

To continue:

the fifth root would be a function

the sixth might or might not

the seventh would be

the eighth might or might not

and so on. The pattern should be clear.

They are all functions.

Diane M had a good point, but it is actually moot. Even-numbered roots (square root, 4th root, 6th root, etc.) are ALWAYS positive by DEFINITION. Therefore, there is NO ambiguity associated with them, and they are ALWAYS functions.

When a function is defined by a formula, it is always going to satisfy the meaning of a function unless it contains one of the following "red flags":

The word "OR"

A "+/-" sign (plus OR minus)

y^2 (or y to any even power)

|y|

The last two are not single-valued because solving them involves placing a "plus OR minus" sign in front of x.

So ...

y = Power of x

y = Root of x

contain none of these red flags and are always single-valued functions!