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The functions we are comparing are \(\sqrt{x}\) and \(\sqrt[3]{x}\). The square root function can be defined as: \[y = \sqrt{x}\] The cube root function can be defined as: \[y = \sqrt[3]{x}\] Both functions have a domain of all non-negative real numbers, that is, \(x \geq 0\). The ranges for both functions are all real numbers, since they can have both positive and negative values.

For \(0<x<1\), let's examine the behavior of each of the functions in that interval: - The square root function yields a result between 0 and 1 as the square root of a number between 0 and 1 will always be greater than that number (e.g., \(\sqrt{0.5}\approx0.71\)). - The cube root function also yields a result between 0 and 1. However, the cube root of a number between 0 and 1 will be less than the square root (e.g., \(\sqrt[3]{0.5}\approx0.79\)). Putting the observations together, we conclude \(\sqrt{x}<\sqrt[3]{x}\) when \(0<x<1\).

Now let's examine the behavior of each of the functions for \(x>1\): - The square root function yields a result greater than 1 as the square root of a number greater than 1 will also be greater than 1 (e.g., \(\sqrt{2}\approx1.41\)). - The cube root function also yields a result greater than 1. However, the cube root of a number greater than 1 will be less than the square root (e.g., \(\sqrt[3]{2}\approx1.26\)). Putting the observations together, we conclude \(\sqrt{x}>\sqrt[3]{x}\) when \(x>1\).

To sketch the graphs of the functions, recall their properties and compare their behavior on the specified interval. 1. Draw the x-axis and y-axis. 2. Graph the square root function (a curving line starting at the origin, increasing, and becoming less steep as it goes on). 3. Graph the cube root function (a curving line starting at negative infinity passing through the origin, and increasing at a slower rate than the square root function as x increases). Both graphs will intersect at two points: (0,0) and (1,1). On the interval \(0 < x < 1\), the graph of \(\sqrt{x}\) will lie below the graph of \(\sqrt[3]{x}\). In contrast, for \(x > 1\), the graph of \(\sqrt{x}\) will lie above the graph of \(\sqrt[3]{x}\).