1. **Problem statement:** We are given two functions: $$f_9(w) = w^2 + 2$$ $$g_9(v) = 3v^2 + 6v - 6$$ We need to find the composition $$f_9(g_9(x))$$, which means substituting $$g_9(x)$$ into $$f_9$$. 2. **Formula and rules:** The composition of functions is defined as: $$f(g(x)) = f\bigl(g(x)\bigr)$$ This means wherever there is a $$w$$ in $$f_9(w)$$, replace it with $$g_9(x)$$. 3. **Substitute:** $$f_9(g_9(x)) = \bigl(g_9(x)\bigr)^2 + 2$$ Substitute $$g_9(x) = 3x^2 + 6x - 6$$: $$= \left(3x^2 + 6x - 6\right)^2 + 2$$ 4. **Expand the square:** Use the formula $$(a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc$$ with $$a=3x^2$$, $$b=6x$$, $$c=-6$$: $$= (3x^2)^2 + (6x)^2 + (-6)^2 + 2(3x^2)(6x) + 2(3x^2)(-6) + 2(6x)(-6) + 2$$ Calculate each term: $$= 9x^4 + 36x^2 + 36 + 36x^3 - 36x^2 - 72x + 2$$ 5. **Combine like terms:** $$9x^4 + 36x^3 + (36x^2 - 36x^2) - 72x + (36 + 2)$$ Simplify: $$9x^4 + 36x^3 - 72x + 38$$ 6. **Final expression:** $$f_9(g_9(x)) = 9x^4 + 36x^3 - 72x + 38$$ 7. **Evaluate at points:** - At $$x = -1$$: $$9(-1)^4 + 36(-1)^3 - 72(-1) + 38 = 9(1) - 36 + 72 + 38 = 83$$ - At $$x = 0$$: $$9(0)^4 + 36(0)^3 - 72(0) + 38 = 38$$ - At $$x = 2$$: $$9(2)^4 + 36(2)^3 - 72(2) + 38 = 9(16) + 36(8) - 144 + 38 = 144 + 288 - 144 + 38 = 326$$ 8. **Interpretation:** You can plot these points on a coordinate system with axes $$y$$ and $$z$$, where $$x$$ values are on the number line and the corresponding $$f_9(g_9(x))$$ values are the points in the $$y-z$$ plane.