Form the composites of the following two functions.
f(x)=x2 +7x--4 and g(x)=3x--12
5. (f g)(x)
6. (g g)(x)
thanks!
To form the composite of two functions, follow these steps:
1. Start with the given functions f(x) and g(x).
2. Replace the variable x in the first function with the second function g(x).
3. Simplify the expression obtained after substitution.
Let's apply these steps to find the composites of the given functions:
1. (f ◦ g)(x):
Replace x in f(x) with g(x):
f(g(x)) = (g(x))^2 + 7(g(x)) - 4
= (3x - 12)^2 + 7(3x - 12) - 4
2. (g ◦ g)(x):
Replace x in g(x) with g(x):
g(g(x)) = 3(g(x)) - 12
= 3(3x - 12) - 12
Now, simplify the expressions:
1. (f ◦ g)(x):
= (3x - 12)^2 + 7(3x - 12) - 4
= 9x^2 - 72x + 144 + 21x - 84 - 4
= 9x^2 - 51x + 56
2. (g ◦ g)(x):
= 3(3x - 12) - 12
= 9x - 36 - 12
= 9x - 48
Therefore, the composites are:
5. (f ◦ g)(x) = 9x^2 - 51x + 56
6. (g ◦ g)(x) = 9x - 48