Recall that:
[tex](h\circ l)(x)=h(l(x)).[/tex]
Therefore, to find the composite functions, first, we evaluate f(x) at:
[tex]x^2-3[/tex]
and get:
[tex]f(x^2-3)=4(x^2-3)^2+1.[/tex]
Simplifying the above result, we get:
[tex]f(g(x))=4(x^4-6x^2+9)+1=4x^4-24x^2+37.[/tex]
Now, we evaluate g(x) at:
[tex]4x^2+1[/tex]
and get:
[tex]g(4x^2+1)=(4x^2+1)^2-3.[/tex]
Simplifying the above result, we get:
[tex]g(f(x))=16x^4+8x^2-2.[/tex]
Answer:
[tex]\begin{gathered} f(g(x))=4x^4-24x^2+37, \\ g(f(x))=16x^4+8x^2-2. \end{gathered}[/tex]
