Given f(x)=x^2 + x + 4 and h(x)= 3x +2, evaluate the composite function.
f [h(x)]
f(h(x) )
= f(3x+2)
= (3x+2)^2 + (3x+2) + 4
= 9x^2 + 12x + 4 + 3x + 2 + 4
= 9x^2 + 15x + 10
To evaluate the composite function f[h(x)], we need to substitute the function h(x) into the function f(x) and simplify.
First, we substitute h(x) into f(x):
f[h(x)] = (h(x))^2 + (h(x)) + 4
Next, we substitute the expression for h(x):
f[h(x)] = (3x + 2)^2 + (3x + 2) + 4
Simplifying further, we expand the squared term:
f[h(x)] = (9x^2 + 12x + 4) + (3x + 2) + 4
Combine like terms:
f[h(x)] = 9x^2 + 15x + 10
So, the composite function f[h(x)] is equal to 9x^2 + 15x + 10.
To evaluate the composite function f[h(x)], we need to substitute the expression for h(x) into f(x).
Step 1: Replace x in f(x) with h(x):
f[h(x)] = (3x + 2)^2 + (3x + 2) + 4
Step 2: Simplify the expression:
f[h(x)] = (9x^2 + 12x + 4) + (3x + 2) + 4
Step 3: Combine like terms:
f[h(x)] = 9x^2 + 12x + 4 + 3x + 2 + 4
Step 4: Simplify once more:
f[h(x)] = 9x^2 + 15x + 10
So, f[h(x)] = 9x^2 + 15x + 10.