If \( g(x)=2 x-4\{x \in R: x \geq 0\} \) a) Calculate \( g^{-1}(x) \) b) Sketch the graphs of both functions on the same set of axes c) What is the relationship between the graphs?

It seems that I am unable to plot the graphs directly. However, I can guide you on how to sketch them manually. ### Sketching the Graphs 1. **Graph of \( g(x) = 2x - 4 \)**: - Start at the point \( (0, -4) \). - From there, use the slope of 2 to find another point. For example, when \( x = 2 \): \[ g(2) = 2(2) - 4 = 0 \quad \text{(point (2, 0))} \] - Continue plotting points, such as \( (4, 4) \) and \( (6, 8) \). - Draw a straight line through these points, extending to the right since \( g(x) \) is defined for \( x \geq 0 \). 2. **Graph of \( g^{-1}(x) = \frac{x + 4}{2} \)**: - Start at the point \( (0, 2) \). - Use the slope of \( \frac{1}{2} \) to find another point. For example, when \( x = 0 \): \[ g^{-1}(0) = \frac{0 + 4}{2} = 2 \quad \text{(point (0, 2))} \] - When \( x = 2 \): \[ g^{-1}(2) = \frac{2 + 4}{2} = 3 \quad \text{(point (2, 3))} \] - Continue plotting points, such as \( (4, 4) \) and \( (6, 5) \). - Draw a straight line through these points, extending to the left since \( g^{-1}(x) \) is defined for all \( x \). ### Relationship Between the Graphs - The line \( y = x \) can be drawn as a dashed line to show the reflection. - The points where the two graphs intersect (like \( (4, 4) \)) confirm that they are inverses of each other. ### Summary - The inverse function \( g^{-1}(x) = \frac{x + 4}{2} \) has been calculated. - The graphs of \( g(x) \) and \( g^{-1}(x) \) are reflections across the line \( y = x \). - You can sketch these graphs on graph paper or using graphing software for a visual representation.