Questions: For f(x)=1-x and g(x)=4x^2+x+3, find the following functions. a. (f∘g)(x); b. (g∘f)(x); c. (f∘g)(3); d. (g∘f)(3) a. (f∘g)(x)=

Solution Steps

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:

$$= -4(9) - 3 - 2 = -36 - 3 - 2 = -41$$

Final Answer