Answer
a. $6x - 3x^2 - 1$ Domain: (-∞, ∞)
b. $ -9x^2 + 12x - 3$ Domain: (-∞, ∞)
c. $9x -4$ Domain: (-∞, ∞)
d. $ 4x - 6x^2 + 4x^3 - x^4$ Domain: (-∞, ∞)
Work Step by Step
Given $f(x) = 3x-1$ and $g(x)= 2x-x^2$, we need to find the following and their respective domains:
a. f o g
$f(2x-x^2) = 3(2x-x^2) - 1 = 6x - 3x^2 - 1$
Domain: (-∞, ∞)
b. g o f
$g(3x-1) = 2(3x-1) - (3x-1)^2 = 6x - 2 - (9x^2 - 6x + 1) = -9x^2 + 12x - 3$
Domain: (-∞, ∞)
c. f o f
$f ( 3x-1) = 3(3x-1) - 1 = 9x -4$
Domain: (-∞, ∞)
d. g o g
$g ( 2x - x^2) = 2(2x - x^2) - (2x - x^2) ^ 2 = 4x - 2x^2 - x^4 + 4x^3 - 4x^2$
$= 4x - 6x^2 + 4x^3 - x^4$
Domain: (-∞, ∞)
