ANSWER
The vertex of (f o g)(x) is on the x-axis while the vertex of (g o f)(x) is on the y-axis
EXPLANATION
First, we have to find the composite functions. In this notation, this means that we have to substitute x in the first function with the second function,
[tex](f\circ g)(x)=f(g(x))=(g(x))^2=(3x-2)^2[/tex]
Let's expand this function using the binomial squared rule,
[tex](3x-2)^2=(3x)^2-2\cdot3x\cdot2+2^2=9x^2-12x+4[/tex]
For the second composition, we have,
[tex](g\circ f)(x)=g(f(x))=3f(x)-2=3x^2-2[/tex]
Both composite functions are written in standard form,
[tex]y=ax^2+bx+c[/tex]
The x-coordinate of the vertex is given by,
[tex]x_v=\frac{-b}{2a}[/tex]
For the first composite function, a = 9, and b = -12,
[tex]x_v=\frac{-(-12)}{2\cdot9}=\frac{12}{18}=\frac{2}{3}[/tex]
And the y-coordinate of the vertex is,
[tex]y_v=9x_v^2-12x_v+4=9\left(\frac{2}{3}\right)^2-12\left(\frac{2}{3}\right)+4=0[/tex]
So, the vertex of the first composite function is at the point (2/3, 0).
For the second composite function, a = 3, and b = 0. This means that the x-coordinate of the vertex is 0 and the y-coordinate is,
[tex]y_v=3\cdot0-2=-2[/tex]
So, the vertex of the second composite function is at the point (0, -2).
Therefore, the vertices are located on each axis: the vertex of (f o g)(x) is on the x-axis while the vertex of (g o f)(x) is on the y-axis.
