Question

1. investigate the range of each parent - function below over the interval 0 < x < 1. then order the ranges from least to greatest. identity function, reciprocal function, quadratic function, cubic function, square root function.

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.