Final answer:
To find the compositions (fog)(x) and (gof)(x), we substitute g(x) into f(x), and f(x) into g(x), respectively. For the given functions, (fog)(x) and (gof)(x) can be written as (x^2 - 9)^2 + 9 and (x^2 + 9)^2 - 9. The specific values at x=3 are (fog)(3) = 9 and (gof)(3) = 315.
Step-by-step explanation:
To find (fog)(x) and (gof)(x), we have to perform function composition.
This means we will substitute the function g(x) into f(x) for (fog)(x), and substitute f(x) into g(x) for (gof)(x).
For (fog)(x), we substitute g(x) into f(x), so we get:
f(g(x)) = f(x^2 - 9) = (x^2 - 9)^2 + 9.
For (gof)(x), we substitute f(x) into g(x), so we get:
g(f(x)) = g(x^2 + 9) = (x^2 + 9)^2 - 9.
To find (fog)(3) and (gof)(3), we substitute 3 into the respective compositions:
- (fog)(3) = f(g(3)) = f(3^2 - 9) = f(0) = 0^2 + 9 = 9.
- (gof)(3) = g(f(3)) = g(3^2 + 9) = g(18) = 18^2 - 9 = 324 - 9 = 315.