Answer:
1. f(g(4)) - g(f(4)) = -6
2. f(f(x)) = 125x^4 - 250x^2 + 120
3. a + b = 0
4. g(4) = -9
Step-by-step explanation:
1.
First we need to find f(4) and g(4):
f(4) = 4 + 3 = 7
g(4) = 3*4 + 5 = 17
Then, we find g(f(4)) = g(7):
g(7) = 3*7 + 5 = 26
And we find f(g(4)) = f(17):
f(17) = 17 + 3 = 20
so f(g(4)) - g(f(4)) = 20 - 26 = -6
2.
To find f(f(x)), we use the value of f(x) for every x in f(x):
f(f(x)) = 5*(f(x))^2 - 5 = 5*(5x^2 - 5)^2 - 5 = 5*(25x^4 - 50x^2 + 25) - 5
f(f(x)) = 125x^4 - 250x^2 + 120
3.
To find f(g(x)), we use the value of g(x) for every x in f(x):
f(g(x)) = g(x) + 6 = ax + b + 6 = 3x + 3
ax + (b+6) = 3x + 3 -> a = 3 and b = -3
a + b = 3 - 3 = 0
4.
If we assume g(x) = ax + b, we have:
g(f(x)) = a*(2x - 3) + b = 2ax - 3a + b = 5 - 4x
2a = -4 -> a = -2
-3a + b = 5
6 + b = 5 -> b = -1
g(x) = -2x - 1
g(4) = -2*4 - 1 = -9
