Question

given the functions f(x) = x^2 and g(x) = 3x-2, determine how the vertex of the composite function would differ between (fg)(x) and (gf)(x)

Answer 1

The vertex of
`f\:\circ \:\:g(x)=((2)/(3),0)`

The vertex of
`g\:\circ \:\:f(x)=(0,-2)`

Explanation:

Given : Functions
`f(x) = x^2` and
`g(x) = 3x-2`

To determine : How the vertex of the composite function would differ between (fg)(x) and (gf)(x)

Solution : First we find the composite function

`f(x) = x^2` and
`g(x) = 3x-2`

1)
`f\:\circ \:\:g(x)`

For
`f(x) = x^2` substitute x with
`g(x) = 3x-2`

`f\:\circ \:\:g(x)=(3x-2)^2`

`f\:\circ \:\:g(x)=9(x-(2)/(3))^2`

Vertex form is
`y=a(x-h)^2+k`

Comparing with
`f\:\circ \:\:g(x)`

a=9, vertex
`(h,k)=((2)/(3),0)`

Therefore, The vertex of
`f\:\circ \:\:g(x)=((2)/(3),0)` ........[1]

2)
`g\:\circ \:\:f(x)`

For
`g= 3x-2` substitute x with
`f(x) = x^2`

`g\:\circ \:\:f(x)=3x^2-2`

To find the vertex of a quadratic function
`y=ax^2+bx+c` the vertex is
`(-(b)/(2a),f((b)/(2a)))`

Comparing with
`g\:\circ \:\:f(x)`

a=3 b=0,c=-2 substitute value,

`(-(b)/(2a),f((b)/(2a)))=(0,-2)`

Therefore, The vertex of
`g\:\circ \:\:f(x)=(0,-2)` ...........[2]

Hence, The vertex of the composite function differ between
`f\:\circ \:\:g(x)` and
`g\:\circ \:\:f(x)` by [1] and [2]