Answer:
The values of the composite functions are (f * g)(x) = -30x^9 + 217x^8 - 384x^7 - 1290x^6 + 1846x^5 - 1566x^4 + 655x^3 + 1377x^2 - 330x + 19;
(g * f)(x) = -270x^9 - 3003x^8 + 13122x^7 - 20763x^6 + 17574x^5 - 13230x^4 + 5301x^3 + 9641x^2 - 1122x + 205.
Explanation:
Given:
(f * g)(x) and (g * f)(x)
1) (f * g)(x):
Since we're asked for (f * g)(x), it means we need to find the composition of f and g. This signifies that we need to plug the entire function g(x) as the argument into function f(x).
Step 1: Replace x in f(x) with g(x):
f(x) = 3x^3 + 11x^2 - 12x + 5
f(g(x)) = 3(g(x))^3 + 11(g(x))^2 - 12(g(x)) + 5
Step 2: Expand the expression:
Now, we need to expand the expression by substituting g(x) wherever we see x.
Remember, g(x) is defined as -10x^3 + 7x^2 + 9x - 4.
Replace (g(x))^3 with -10(x^3)^3 + 7(x^2)^3 + 9(x^3) - 4(x)^3
Replace (g(x))^2 with -100(x^3)^2 + 140(x^2)x + 81(x^2)^2 - 36x^2 + 16(x)^2
Replace (g(x)) with -30x^3 + 77x^2 + 81x - 36
f(g(x)) = 3(-10(x^3)^3 + 7(x^2)^3 + 9(x^3) - 4(x)^3) + 11(-100(x^3)^2 + 140(x^2)x + 81(x^2)^2 - 36x^2 + 16(x)^2) +
-12(-30x^3 + 77x^2 + 81x - 36) + 5
Step 3: Simplify the expression:
Combining like terms, we get the simplified expression for (f * g)(x):
(f * g)(x) = -30x^9 + 217x^8 - 384x^7 - 1290x^6 + 1846x^5 - 1566x^4 + 655x^3 + 1377x^2 - 330x + 19
2)
(g * f)(x):
Following the same process, we can find (g * f)(x) by plugging f(x) as the argument into g(x).
Step 1: Replace x in g(x) with f(x):
g(x) = -10x^3 + 7x^2 + 9x - 4
g(f(x)) = -10(f(x))^3 + 7(f(x))^2 + 9(f(x)) - 4
Step 2: Expand the expression:
Replace f(x) with its definition: 3x^3 + 11x^2 - 12x + 5.
Replace (f(x))^3 with 27x^9 + 333x^8 - 1458x^7 + 2307x^6 - 1953x^5
Replace (f(x))^2 with 9x^6 + 222x^5 - 480x^4 + 660x^3 - 240x^2 + 121x
Replace (f(x)) with 9x^3 + 110x^2 - 132x + 45
g(f(x)) = -10(27x^9 + 333x^8 - 1458x^7 + 2307x^6 - 1953x^5) + 7(9x^6 + 222x^5 - 480x^4 + 660x^3 - 240x^2 + 121x) +
-9(9x^3 + 110x^2 - 132x + 45) - 4
Thus,
(f * g)(x) = -30x^9 + 217x^8 - 384x^7 - 1290x^6 + 1846x^5 - 1566x^4 + 655x^3 + 1377x^2 - 330x + 19;
(g * f)(x) = -270x^9 - 3003x^8 + 13122x^7 - 20763x^6 + 17574x^5 - 13230x^4 + 5301x^3 + 9641x^2 - 1122x + 205.