To find f(g(f(g(fg)))) and fg, we first need to clearly define our functions:
- f(x) = 4x
- g(x) = x² + 3x
Let’s start with fg:
fg means f(g(x)). We first calculate g(x):
- g(x) = x² + 3x
Now plug g(x) into f(x):
- f(g(x)) = f(x² + 3x) = 4(x² + 3x) = 4x² + 12x
So, fg = 4x² + 12x.
Next, we determine the domains:
The domain of f(x) = 4x is all real numbers since it’s a linear function.
The domain of g(x) = x² + 3x is also all real numbers as it’s a polynomial.
Thus, the domain of fg = 4x² + 12x is also all real numbers (Domain: ℝ).
Now, for f(g(f(g(fg)))):
We start from fg and work our way back up through the functions:
- fg = 4x² + 12x
- Now apply g: g(fg) = g(4x² + 12x) = (4x² + 12x)² + 3(4x² + 12x)
- Calculate (4x² + 12x)²:
- (4x² + 12x)² = 16x⁴ + 96x³ + 144x²
- Then, 3(4x² + 12x) = 12x² + 36x
- Combining gives us: g(fg) = 16x⁴ + 96x³ + 144x² + 12x² + 36x = 16x⁴ + 96x³ + 156x² + 36x.
Now we find f(g(fg)): f(g(fg)) = f(16x⁴ + 96x³ + 156x² + 36x) = 4(16x⁴ + 96x³ + 156x² + 36x) = 64x⁴ + 384x³ + 624x² + 144x.
Finally, we apply f one more time to get f(g(f(g(fg)))):
Since we already calculated the last function, we can conclude:
- f(g(f(g(fg)))) = 64x⁴ + 384x³ + 624x² + 144x
Thus, the final output is:
- fg = 4x² + 12x
- f(g(f(g(fg)))) = 64x⁴ + 384x³ + 624x² + 144x
The common domain for all functions is all real numbers (Domain: ℝ).