2. For each of the following power function sketch its graphs and state is properties (domain, range, \( \mathrm{x} \& \mathrm{y} y \) intercepts and generalize for the case of each function based on its power and coefficients) \( \begin{array}{llll}\text { a. } f(x)= \pm 2 x^{\frac{3}{2}} & \text { c. } f(x)= \pm 3 x^{ \pm \frac{3}{2}} & \text { e. } f(x)=\frac{5}{x^{-\frac{2}{3}}} & \text { g. } h(x)=\sqrt{2} x^{\frac{4}{3}}\end{array} \)

Let's analyze each of the given power functions one by one, sketch their graphs, and state their properties, including domain, range, \(x\) and \(y\) intercepts. ### a. \( f(x) = \pm 2 x^{\frac{3}{2}} \) **Graph:** - The graph of \( f(x) = 2 x^{\frac{3}{2}} \) is a curve that starts at the origin (0,0) and rises steeply as \( x \) increases. The graph of \( f(x) = -2 x^{\frac{3}{2}} \) is a reflection of the first graph across the x-axis. **Properties:** - **Domain:** \( x \geq 0 \) (since \( x^{\frac{3}{2}} \) is only defined for non-negative \( x \)) - **Range:** \( (-\infty, 0] \) for \( f(x) = -2 x^{\frac{3}{2}} \) and \( [0, \infty) \) for \( f(x) = 2 x^{\frac{3}{2}} \) - **\( x \)-intercept:** \( (0, 0) \) - **\( y \)-intercept:** \( (0, 0) \) ### c. \( f(x) = \pm 3 x^{-\frac{3}{2}} \) **Graph:** - The graph of \( f(x) = 3 x^{-\frac{3}{2}} \) is a hyperbola that approaches the x-axis as \( x \) increases and goes to infinity as \( x \) approaches 0 from the right. The graph of \( f(x) = -3 x^{-\frac{3}{2}} \) is a reflection across the x-axis. **Properties:** - **Domain:** \( x > 0 \) (since \( x^{-\frac{3}{2}} \) is only defined for positive \( x \)) - **Range:** \( (0, \infty) \) for \( f(x) = 3 x^{-\frac{3}{2}} \) and \( (-\infty, 0) \) for \( f(x) = -3 x^{-\frac{3}{2}} \) - **\( x \)-intercept:** None (the function does not cross the x-axis) - **\( y \)-intercept:** None (the function is not defined at \( x = 0 \)) ### e. \( f(x) = \frac{5}{x^{-\frac{2}{3}}} \) **Simplification:** - This can be rewritten as \( f(x) = 5 x^{\frac{2}{3}} \). **Graph:** - The graph of \( f(x) = 5 x^{\frac{2}{3}} \) is similar to that of \( f(x) = 2 x^{\frac{3}{2}} \) but grows less steeply. **Properties:** - **Domain:** \( x \geq 0 \) - **Range:** \( [0, \infty) \) - **\( x \)-intercept:** \( (0, 0) \) - **\( y \)-intercept:** \( (0, 0) \) ### g. \( h(x) = \sqrt{2} x^{\frac{4}{3}} \) **Graph:** - The graph of \( h(x) = \sqrt{2} x^{\frac{4}{3}} \) is similar to that of \( f(x) = 2 x^{\frac{3}{2}} \) but scaled vertically by \( \sqrt{2} \). **Properties:** - **Domain:** \( x \geq 0 \) - **Range:** \( [0, \infty) \) - **\( x \)-intercept:** \( (0, 0) \) - **\( y \)-intercept:** \( (0, 0) \) ### Generalization Based on Power and Coefficients 1. **For functions of the form \( f(x) = k x^n \) where \( n \) is a positive rational number:** - **Domain:** \( x \geq 0 \) - **Range:** \( [0, \infty) \) if \( k > 0 \) and \( (-\infty, 0] \) if \( k < 0 \) - **Intercepts:** \( (0, 0) \) for both \( x \) and \( y \) intercepts. 2. **For functions of the form \( f(x) = k x^{-n} \) where \( n \) is a positive rational number:** - **Domain:** \( x > 0 \) - **Range:** \( (0, \infty) \) if \( k > 0 \) and \( (-\infty, 0) \) if \( k < 0 \) - **Intercepts:** No \( x \)-intercept or \( y \)-intercept. This analysis provides a comprehensive understanding of the properties of the given power functions.