4. investigate the range of each parent - function below over the interval 0 < x < 1. then order the ranges…

Explanation:

Step1: Define the parent - functions

The identity function is (y = x), the reciprocal function is (y=\frac{1}{x}), the quadratic function is (y = x^{2}), the cubic function is (y=x^{3}), and the square - root function is (y = \sqrt{x}).

Step2: Find the range of the identity function on (0\lt x\lt1)

For (y = x), when (x\in(0,1)), the range is ((0,1)).

Step3: Find the range of the reciprocal function on (0\lt x\lt1)

For (y=\frac{1}{x}), as (x) approaches (0) from the right, (y) approaches (+\infty), and as (x) approaches (1), (y = 1). So the range is ((1,+\infty)).

Step4: Find the range of the quadratic function on (0\lt x\lt1)

For (y = x^{2}), when (x = 0), (y = 0) (not in the open - interval), when (x = 1), (y = 1) (not in the open - interval), and the function is increasing on ((0,1)). So the range is ((0,1)). But for any (x\in(0,1)), (x^{2}\lt x).

Step5: Find the range of the cubic function on (0\lt x\lt1)

For (y=x^{3}), when (x = 0), (y = 0) (not in the open - interval), when (x = 1), (y = 1) (not in the open - interval), and the function is increasing on ((0,1)). Also, for any (x\in(0,1)), (x^{3}\lt x^{2}). So the range is ((0,1)) and (x^{3}\lt x^{2}\lt x).

Step6: Find the range of the square - root function on (0\lt x\lt1)

For (y=\sqrt{x}), when (x = 0), (y = 0) (not in the open - interval), when (x = 1), (y = 1) (not in the open - interval), and the function is increasing on ((0,1)). Also, for any (x\in(0,1)), (x\lt\sqrt{x}).

Step7: Order the ranges from least to greatest

The order of the ranges from least to greatest is: cubic function ((y = x^{3})), quadratic function ((y = x^{2})), identity function ((y = x)), square - root function ((y=\sqrt{x})), reciprocal function ((y=\frac{1}{x})).

Answer:

Cubic function, Quadratic function, Identity function, Square - root function, Reciprocal function.