- (fof)(x) = x⁴ + 4x³ + 13x² + 4x + 1
These calculations show how the order of operations within composite functions affects the final result.
Given functions f(x) = x²+x and g(x) = x + 8, let's evaluate the composite functions:
1. (f°g)(x):
This means we evaluate g(x) first, then plug the result into f(x). So, (f°g)(x) = f(g(x)) = f(x+8) = (x+8)² + (x+8) = x² + 20x + 80.
2. (gof)(x):
Similarly, we evaluate f(x) first, then plug the result into g(x). So, (gof)(x) = g(f(x)) = g(x²+x) = (x²+x) + 8 = x² + x + 8.
3. (fof)(x):
This involves nesting f within itself. We evaluate f(x) twice. So, (fof)(x) = f(f(x)) = f(x²+x) = ((x²+x)² + (x²+x)) = x⁴ + 4x³ + 13x² + 4x + 1.
4. (g°g)(x):
Similar to (fof)(x), we evaluate g(x) twice. So, (g°g)(x) = g(g(x)) = g(x+8) = (x+8) + 8 = x + 16.