Hey Phiwo🥺❤️, good to see you again. a) Determine the equations of the function and the inverse function. Step 1: Determine the equation of the function from the graph. The graph shows a straight line passing through the origin $(0,0)$ and the point $(3,1)$. The general equation for a straight line is $y = mx + c$, where $m$ is the slope and $c$ is the y-intercept. Calculate the slope $m$: $$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - 0}{3 - 0} = \frac{1}{3}$$ Since the line passes through $(0,0)$, the y-intercept $c$ is $0$. Therefore, the equation of the function is: $$y = \frac{1}{3}x$$ Step 2: Determine the equation of the inverse function. To find the inverse function, swap $x$ and $y$ in the original function's equation and then solve for $y$. Original function: $y = \frac{1}{3}x$ Swap $x$ and $y$: $x = \frac{1}{3}y$ Multiply both sides by 3 to solve for $y$: $$3x = y$$ So, the equation of the inverse function is: $$y = 3x$$ The equations are: Function: $\boxed{y = \frac{1}{3}x}$ Inverse function: $\boxed{y = 3x}$ b) Sketch the graphs of the function and its inverse on the same set of axes. The graph of the function $y = \frac{1}{3}x$ is already provided in Figure 4.18. To sketch the inverse function $y = 3x$ on the same axes: • The inverse function $y = 3x$ is also a straight line that passes through the origin $(0,0)$. • To find another point on the inverse graph, substitute $x=1$ into $y=3x$, which gives $y=3$. So, the point $(1,3)$ is on the inverse graph. • Similarly, if $x=-1$, then $y=3(-1)=-3$. So, the point $(-1,-3)$ is on the inverse graph. • Draw a straight line connecting these points. This line will be a reflection of the original function $y = \frac{1}{3}x$ across the line $y=x$. Send me the next one 📸