Respuesta :
We are given the following functions:
[tex]\begin{gathered} f\mleft(x\mright)=x^2+2x \\ g\mleft(x\mright)=x+3 \end{gathered}[/tex]We are asked to determine the following composition:
[tex](f\circ g)(x)=f(g(x))[/tex]This means that where we have "x" in f we will replace it for the function g, like this:
[tex]f(g(x))=(x+3)^2+2(x+3)[/tex]Simplifying we get:
[tex]\begin{gathered} f(g(x))=x^2+6x+9+2x+6 \\ f(g(x))=x^2+8x+15 \end{gathered}[/tex]Now we are asked to determine the following composition:
[tex](g\circ f)(x)=g(f(x))[/tex]This means that where there is "x" in g we will replace it by f:
[tex]\begin{gathered} g(f(x))=(x^2+2x)+3 \\ g(f(x))=x^2+2x+3 \end{gathered}[/tex]Now we are asked to determine:
[tex](f\circ f)(x)=f(f(x))[/tex]Replacing the value of f in f:
[tex]f(f(x))=(x^2+2x)^2+2(x^2+2x)[/tex]Simplifying:
[tex]\begin{gathered} f(f(x))=x^4+4x^3+4x^2+2x^2+4x \\ f(f(x))=x^4+4x^3+6x^2+4x \end{gathered}[/tex]Finally, we are asked to determine the following composition:
[tex](g\circ g)(x)=g(g(x))[/tex]Replacing we get:
[tex]\begin{gathered} g(g(x))=(x+3)+3 \\ g(g(x))=x+6 \end{gathered}[/tex]