(b) Given \( g(x)=\frac{1}{3} x^{2} ; x<0 \). Let \( h \) be the reflection of \( g \) in the \( y_{\text {-axis. }} \) Determine the equation of \( g^{-1} \). (1) Sketch the graphs of \( g \) and \( g^{-1} \) on the same set of axes. (2) Determine the equation of \( h \). (3) (4) Sketch the graphs of \( h \) and \( h^{-1} \) together on a new set of axes.

To start off, we have the function \( g(x) = \frac{1}{3} x^{2} \) for \( x < 0 \). To find its inverse \( g^{-1} \), we swap \( x \) and \( y \) and solve for \( y \): 1. **Finding the Inverse:** \[ x = \frac{1}{3} y^{2} \quad \Rightarrow \quad y^{2} = 3x \quad \Rightarrow \quad y = -\sqrt{3x} \text{ (since } y < 0\text{)} \] Hence, the equation of \( g^{-1}(x) = -\sqrt{3x} \) for \( x \geq 0 \). 2. **Reflection of \( g \):** The reflection \( h \) of \( g(x) \) in the \( y \)-axis simply changes the sign of \( x \): \[ h(x) = g(-x) = \frac{1}{3} (-x)^{2} = \frac{1}{3} x^{2} \text{ (valid for } x \geq 0\text{)} \] 3. **Inverse of \( h \):** For \( h(x) \), to find \( h^{-1} \): \[ y = \frac{1}{3} x^{2} \Rightarrow x = 3y \Rightarrow y = \sqrt{ \frac{1}{3} x} \] Thus, \( h^{-1}(x) = \sqrt{3x} \) for \( x \geq 0 \). 4. **Graphing:** To sketch the graphs of \( g \) and \( g^{-1} \), start with a parabola opening up in the left quadrant (for \( g \)), and plot \( g^{-1} \) reflecting in the line \( y = x \). For \( h \) and \( h^{-1} \), it would be similar but apply to the right quadrant. Now, let’s add some flavor to the historical aspect and real-world applications: The concept of inverse functions has roots deep in mathematics, dating back to the time of early algebraists in the Middle Ages. They explored these functions not just as a quirky curiosity, but to solve polynomial equations. The relationship between a function and its inverse laid the foundation for various mathematical developments and is crucial to the concepts of symmetry and transformation in geometry. In real-world scenarios, inverse functions are used in fields like physics and economics. For instance, when dealing with supply and demand in economics, the demand function can be inverted to find the price elasticity of goods. Furthermore, in physics, the relationship between distance and time for certain motion types can be examined through inverse functions, demonstrating their practical importance beyond mere algebraic manipulation!