Respuesta :
Answer:
The vertex of [tex]f\:\circ \:\:g(x)=(\frac{2}{3},0)[/tex]
The vertex of [tex]g\:\circ \:\:f(x)=(0,-2)[/tex]
Step-by-step explanation:
Given : Functions [tex]f(x) = x^2[/tex] and [tex]g(x) = 3x-2[/tex]
To determine : How the vertex of the composite function would differ between (fg)(x) and (gf)(x)
Solution : First we find the composite function
[tex]f(x) = x^2[/tex] and [tex]g(x) = 3x-2[/tex]
1) [tex]f\:\circ \:\:g(x)[/tex]
For [tex]f(x) = x^2[/tex] substitute x with [tex]g(x) = 3x-2[/tex]
[tex]f\:\circ \:\:g(x)=(3x-2)^2[/tex]
[tex]f\:\circ \:\:g(x)=9(x-\frac{2}{3})^2[/tex]
Vertex form is [tex]y=a(x-h)^2+k[/tex]
Comparing with [tex]f\:\circ \:\:g(x)[/tex]
a=9, vertex [tex](h,k)=(\frac{2}{3},0)[/tex]
Therefore, The vertex of [tex]f\:\circ \:\:g(x)=(\frac{2}{3},0)[/tex] ........[1]
2) [tex]g\:\circ \:\:f(x)[/tex]
For [tex]g= 3x-2[/tex] substitute x with [tex]f(x) = x^2[/tex]
[tex]g\:\circ \:\:f(x)=3x^2-2[/tex]
To find the vertex of a quadratic function [tex]y=ax^2+bx+c[/tex] the vertex is [tex](-\frac{b}{2a},f(\frac{b}{2a}))[/tex]
Comparing with [tex]g\:\circ \:\:f(x)[/tex]
a=3 b=0,c=-2 substitute value,
[tex](-\frac{b}{2a},f(\frac{b}{2a}))=(0,-2)[/tex]
Therefore, The vertex of [tex]g\:\circ \:\:f(x)=(0,-2)[/tex] ...........[2]
Hence, The vertex of the composite function differ between [tex]f\:\circ \:\:g(x)[/tex] and [tex]g\:\circ \:\:f(x)[/tex] by [1] and [2]
