To solve the given problem, we need to find the compositions of the functions \( f(x) = 1 - x \) and \( g(x) = 4x^2 + x + 3 \).
a. For \( (f \circ g)(x) \), substitute \( g(x) \) into \( f(x) \).
b. For \( (g \circ f)(x) \), substitute \( f(x) \) into \( g(x) \).
c. For \( (f \circ g)(3) \), first find \( g(3) \) and then substitute the result into \( f(x) \).
d. For \( (g \circ f)(3) \), first find \( f(3) \) and then substitute the result into \( g(x) \).
The composition of functions involves applying one function to the results of another. For two functions \( f(x) \) and \( g(x) \), the composition \( (f \circ g)(x) \) means \( f(g(x)) \), and \( (g \circ f)(x) \) means \( g(f(x)) \).
Given:
- \( f(x) = 1 - x \)
- \( g(x) = 4x^2 + x + 3 \)
To find \( (f \circ g)(x) \), substitute \( g(x) \) into \( f(x) \):
\[
(f \circ g)(x) = f(g(x)) = f(4x^2 + x + 3) = 1 - (4x^2 + x + 3)
\]
Simplify the expression:
\[
= 1 - 4x^2 - x - 3 = -4x^2 - x - 2
\]
To find \( (g \circ f)(x) \), substitute \( f(x) \) into \( g(x) \):
\[
(g \circ f)(x) = g(f(x)) = g(1 - x) = 4(1 - x)^2 + (1 - x) + 3
\]
First, expand \( (1 - x)^2 \):
\[
(1 - x)^2 = 1 - 2x + x^2
\]
Substitute back into the expression:
\[
= 4(1 - 2x + x^2) + 1 - x + 3
\]
Distribute the 4:
\[
= 4 - 8x + 4x^2 + 1 - x + 3
\]
Combine like terms:
\[
= 4x^2 - 9x + 8
\]
Substitute \( x = 3 \) into \( (f \circ g)(x) = -4x^2 - x - 2 \):
\[
(f \circ g)(3) = -4(3)^2 - 3 - 2
\]
Calculate:
\[
= -4(9) - 3 - 2 = -36 - 3 - 2 = -41
\]
- \( (f \circ g)(x) = \boxed{-4x^2 - x - 2} \)
- \( (g \circ f)(x) = \boxed{4x^2 - 9x + 8} \)
- \( (f \circ g)(3) = \boxed{-41} \)
Answered
