Question

Given the functions f(x) = 3x and g(x) = x^2 + 1, find [gf(x)] - [fg(x)].

Answer 1

To find [gf(x)] - [fg(x)], we first need to find the compositions gf(x) and fg(x). gf(x) = 9x^2 + 1 and fg(x) = 3x^2 + 3. Subtracting fg(x) from gf(x) gives us 6x^2 - 2.

Step-by-step explanation:

To find [gf(x)] - [fg(x)], we first need to find the compositions gf(x) and fg(x).

1. gf(x) = g(f(x))
2. fg(x) = f(g(x))

Let's calculate these compositions:

gf(x) = g(f(x)) = g(3x) = (3x)^2 + 1 = 9x^2 + 1

fg(x) = f(g(x)) = f(x^2 + 1) = 3(x^2 + 1) = 3x^2 + 3

Now, we can subtract [fg(x)] from [gf(x)]:

[gf(x)] - [fg(x)] = (9x^2 + 1) - (3x^2 + 3) = 9x^2 + 1 - 3x^2 - 3 = 6x^2 - 2

Therefore, the expression [gf(x)] - [fg(x)] simplifies to 6x^2 - 2.