Answer: Provided the two functions, f(x) and g(x), we have to find the composite of these two functions at x = - 2:
[tex]\begin{gathered} f(x)=-\frac{1}{2}x^2+5x \\ \\ g(x)=x^2+2 \end{gathered}[/tex]
The composite function is as follows:
[tex]\begin{gathered} f(g(x))=-\frac{1}{2}(x^2+2)^2+5(x^2+2) \\ \\ \\ f(g(x))=-\frac{1}{2}[x^4+4x^2+4]+5x^2+10 \\ \\ \\ f(g(x))=-\frac{x^4}{2}-2x^2-2+5x^2+10 \\ \\ f(g(x))=-\frac{x^4}{2}-2x^2-2+5x^2+10 \\ \\ \\ f(g(x))=-\frac{x^4}{2}+3x^2+8 \\ \\ \\ f(g(-2))=-\frac{(-2)^4}{2}+3(-2)^2+8 \\ \\ \\ f(g(-2))=-\frac{(-2)^4}{2}+3(-2)^2+8=-8+12+8=12 \\ \\ \\ f(g(-2))=12 \end{gathered}[/tex]
The answer is 12.
