Find f(g(f(g(fg)))) and fg and their domains f(x) = x + 6 and g(x) = 5x^2

To find the composition of functions f and g, we start by defining the functions based on the given information:

f(x) = x + 6

g(x) = 5x²

1. **Finding fg (f composed with g):**

To find f(g(x)), we will substitute g(x) into f(x):

f(g(x)) = f(5x²)

Now, we replace the x in f(x) with 5x²:

f(g(x)) = 5x² + 6

So, fg = 5x² + 6.

2. **Finding g(f(x)):**

Now, we need to find g(f(x)). Substitute f(x) into g(x):

g(f(x)) = g(x + 6)

Now replace x in g(x) with (x + 6):

g(f(x)) = 5(x + 6)²

Expanding this:

g(f(x)) = 5(x² + 12x + 36) = 5x² + 60x + 180

g(f(x)) = 5x² + 60x + 180.

3. **Finding f(g(f(g(fg)))):**

We already know that fg = 5x² + 6. Now, we need to find g(fg) first:

g(fg) = g(5x² + 6) = 5(5x² + 6)²

Next, we compute the expression:

g(fg) = 5(25x⁴ + 60x² + 36) = 125x⁴ + 300x² + 180

Now, we find f(g(fg)):

f(g(fg)) = f(125x⁴ + 300x² + 180) = 125x⁴ + 300x² + 180 + 6 = 125x⁴ + 300x² + 186

So, f(g(f(g(fg)))) = 125x⁴ + 300x² + 186.

4. **Domains of the functions:**

For the functions provided:

• The domain of f(x) = x + 6 is all real numbers, denoted as R.
• The domain of g(x) = 5x² is also all real numbers, denoted as R.
• The domain of fg = 5x² + 6 is also all real numbers, denoted as R.
• Finally, the domain of f(g(f(g(fg)))) = 125x⁴ + 300x² + 186 is all real numbers, denoted as R.

In conclusion:

• fg = 5x² + 6
• f(g(f(g(fg)))) = 125x⁴ + 300x² + 186
• Domains: All functions have a domain of all real numbers, R.