Okay, Phiwo🥺❤️, let's solve question 2.4. a) Determine the equations of the function and the inverse function. Step 1: Determine the equation of the inverse function $f^{-1}(x)$ from the graph (Figure 4.21). The graph shows a straight line labeled $f^{-1}$ passing through the points $(0,-2)$ and $(4,0)$. The general equation for a straight line is $y = mx + c$. Calculate the slope $m$: $$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - (-2)}{4 - 0} = \frac{2}{4} = \frac{1}{2}$$ The y-intercept $c$ is the y-coordinate when $x=0$, which is $-2$. Therefore, the equation of the inverse function is: $$f^{-1}(x) = \frac{1}{2}x - 2$$ Step 2: Determine the equation of the original function $f(x)$. To find the original function, swap $x$ and $y$ in the inverse function's equation and then solve for $y$. Let $y = f^{-1}(x)$, so $y = \frac{1}{2}x - 2$. Swap $x$ and $y$: $$x = \frac{1}{2}y - 2$$ Add 2 to both sides: $$x + 2 = \frac{1}{2}y$$ Multiply both sides by 2: $$2(x + 2) = y$$ $$y = 2x + 4$$ So, the equation of the function is: $$f(x) = 2x + 4$$ The equations are: Function: $\boxed{f(x) = 2x + 4}$ Inverse function: $\boxed{f^{-1}(x) = \frac{1}{2}x - 2}$ b) Sketch the graphs of the function and its inverse on the same set of axes. The graph of the inverse function $f^{-1}(x) = \frac{1}{2}x - 2$ is already provided in Figure 4.21. To sketch the original function $f(x) = 2x + 4$ on the same axes: • The function $f(x) = 2x + 4$ is a straight line with a y-intercept of $4$. So, it passes through the point $(0,4)$. • To find the x-intercept, set $y=0$: $0 = 2x + 4 \implies 2x = -4 \implies x = -2$. So, it passes through the point $(-2,0)$. • Draw a straight line connecting these points. This line will be a reflection of the inverse function $f^{-1}(x)$ across the line $y=x$. 3 done, 2 left today. You're making progress.