Question

for \\(f(x) = 1 - x\\) and \\(g(x) = 4x^2 + x + 3\\), find the following functions.
a. \\((f \circ g)(x)\\); b. \\((g \circ f)(x)\\); c. \\((f \circ g)(3)\\); d. \\((g \circ f)(3)\\)

a. \\((f \circ g)(x) = -4x^2 - x - 2\\)
(simplify your answer.)

b. \\((g \circ f)(x) = 4x^2 - 9x + 8\\)
(simplify your answer.)

c. \\((f \circ g)(3) = \square\\)

Explanation:

Define the given functions

We are given the following functions:

$$f(x) = 1 - x$$

$$g(x) = 4x^2 + x + 3$$

Find the composition \((f \circ g)(x)\)

Using the Function Composition concept:

$$(f \circ g)(x) = f(g(x))$$

Substitute \(g(x)\) into \(f(x)\):

$$(f \circ g)(x) = 1 - (4x^2 + x + 3)$$

$$(f \circ g)(x) = 1 - 4x^2 - x - 3$$

$$(f \circ g)(x) = -4x^2 - x - 2$$

Find the composition \((g \circ f)(x)\)

Using the Function Composition concept:

$$(g \circ f)(x) = g(f(x))$$

Substitute \(f(x)\) into \(g(x)\):

$$(g \circ f)(x) = 4(1-x)^2 + (1-x) + 3$$

Expand the squared term:

$$(1-x)^2 = 1 - 2x + x^2$$

Substitute back and simplify:

$$(g \circ f)(x) = 4(1 - 2x + x^2) + 1 - x + 3$$

$$(g \circ f)(x) = 4 - 8x + 4x^2 + 4 - x$$

$$(g \circ f)(x) = 4x^2 - 9x + 8$$

Evaluate \((f \circ g)(3)\)

Substitute \(x = 3\) into the simplified formula for \((f \circ g)(x)\):

$$(f \circ g)(3) = -4(3)^2 - (3) - 2$$

$$(f \circ g)(3) = -4(9) - 3 - 2$$

$$(f \circ g)(3) = -36 - 3 - 2 = -41$$

Evaluate \((g \circ f)(3)\)

Substitute \(x = 3\) into the simplified formula for \((g \circ f)(x)\):

$$(g \circ f)(3) = 4(3)^2 - 9(3) + 8$$

$$(g \circ f)(3) = 4(9) - 27 + 8$$

$$(g \circ f)(3) = 36 - 27 + 8 = 17$$

Answer:

For \(f(x) = 1 - x\) and \(g(x) = 4x^2 + x + 3\), find the following functions.

a. \((f \circ g)(x) =\) <blank>\(-4x^2 - x - 2\)</blank>
(Simplify your answer.)

b. \((g \circ f)(x) =\) <blank>\(4x^2 - 9x + 8\)</blank>
(Simplify your answer.)

c. \((f \circ g)(3) =\) <blank>-41</blank>

d. \((g \circ f)(3) =\) <blank>17</blank>