91Ó°ÊÓ

For both functions, we need to find the domain, range, and any points of interest. Since both functions have a fractional exponent, the domains include all non-negative real numbers: \(x \geq 0\). For both functions, as x increases, the function values also increase, so their ranges are also non-negative real numbers: \(y \geq 0\).

We'll determine key points for both functions within the given interval [0, 81]. Since both functions are continuous and increasing, we can evaluate each function at x = 0, 1, and 81. For \(x^{\frac{1}{4}}\): - At x = 0: \(y = 0^{\frac{1}{4}} = 0\) - At x = 1: \(y = 1^{\frac{1}{4}} = 1\) - At x = 81: \(y = 81^{\frac{1}{4}} = 3\) For \(x^{\frac{1}{5}}\): - At x = 0: \(y = 0^{\frac{1}{5}} = 0\) - At x = 1: \(y = 1^{\frac{1}{5}} = 1\) - At x = 81: \(y = 81^{\frac{1}{5}} = 3\) Thus, both functions share the same key points: (0, 0), (1, 1), and (81, 3).

Notice that \(x^{\frac{1}{4}}\) has a larger exponent than \(x^{\frac{1}{5}}\). This means that as x increases, \(x^{\frac{1}{4}}\) will grow at a faster rate compared to \(x^{\frac{1}{5}}\). In other words, the graph of \(x^{\frac{1}{4}}\) will be steeper compared to the graph of \(x^{\frac{1}{5}}\).

Now that we have all of the necessary information, we can sketch the graphs for each function. 1. Start by plotting the key points for both functions: (0, 0), (1, 1), and (81, 3). 2. Recall that both graphs are increasing and the graph of \(x^{\frac{1}{4}}\) is steeper than the graph of \(x^{\frac{1}{5}}\). Therefore, ensure that the graph for \(x^{\frac{1}{4}}\) appears steeper on your sketch. 3. Connect the key points for each function with smooth curves, ensuring continuity and increasing behavior. The resulting graphs should show that both functions start at the origin, pass through (1, 1), and end at the point (81, 3), with the graph of \(x^{\frac{1}{4}}\) being steeper that the graph of \(x^{\frac{1}{5}}\).