Question

for f(x)=4−x and g(x)=3x²+x+9, find the following functions.
a. (f∘g)(x); b. (g∘f)(x); c. (f∘g)(2); d. (g∘f)(2)
a. (f∘g)(x)=
(simplify your answer.)

Explanation:

Step1: Define composition $(f\circ g)(x)$

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

Step2: Substitute $g(x)$ into $f(x)$

$f(g(x)) = 4 - (3x^2 + x + 9)$

Step3: Simplify the expression

$4 - 3x^2 - x - 9 = -3x^2 - x - 5$

Step4: Define composition $(g\circ f)(x)$

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

Step5: Substitute $f(x)$ into $g(x)$

$g(f(x)) = 3(4 - x)^2 + (4 - x) + 9$

Step6: Expand and simplify

$3(16 - 8x + x^2) + 4 - x + 9 = 48 - 24x + 3x^2 + 4 - x + 9 = 3x^2 -25x +61$

Step7: Calculate $(f\circ g)(2)$

Substitute $x=2$ into $-3x^2 -x -5$:
$-3(2)^2 -2 -5 = -3(4) -7 = -12-7=-19$

Step8: Calculate $(g\circ f)(2)$

Substitute $x=2$ into $3x^2 -25x +61$:
$3(2)^2 -25(2) +61 = 12 -50 +61=23$

Answer:

a. $\boldsymbol{-3x^2 - x - 5}$
b. $\boldsymbol{3x^2 -25x +61}$
c. $\boldsymbol{-19}$
d. $\boldsymbol{23}$