Part A:
Given that [tex]f(x)= \frac{1}{3} x+2[/tex], we find the inverse g(x) as follows:
First, we rewrite the function as: [tex]y=\frac{1}{3} x+2[/tex]
Next, we switch x and y and then solve for y as follows:
[tex]x=\frac{1}{3} y+2 \\ \\ \frac{1}{3} y=x-2 \\ \\ y=3(x-2) \\ \\ y=3x-6[/tex]
We then rewrite the equation using function notation to get:
[tex]g(x)=3x-6[/tex]
Part B:
For two function that are inverse of each other, the composition of the two functions gives x.
Thus,
[tex]f \circ g=f(g(x))=f(3x-6)= \frac{1}{3} (3x-6)+2=x-2+2=x \\ \\ g\circ f=g(f(x))=g( \frac{1}{3} x+2)=3( \frac{1}{3} x+2)-6=x+6-6=x \\ \\ \therefore f\circ g=g\circ f=x[/tex]
Since, f o g = g o f = x, hence g(x) is the inverse of f(x).
Part C:
The graphs of f(x) and g(x) are attached.
From the graph it can be seen that the prperties of both graphs are similar, the properties of one graph in the x-axis correspond to the properties of the other graph in the y-axis and vice-versa. Also, the two lines intersect at a point were x = y.
