To find [tex]\((F-g)(x)\)[/tex], we need to subtract the function [tex]\(g(x)\)[/tex] from the function [tex]\(f(x)\)[/tex].
Given:
[tex]\[ f(x) = 2x^2 + x - 3 \][/tex]
[tex]\[ g(x) = x - 1 \][/tex]
First, let's write out [tex]\((F-g)(x)\)[/tex] explicitly:
[tex]\[ (F-g)(x) = f(x) - g(x) \][/tex]
Now, substitute the expressions for [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[ (F-g)(x) = (2x^2 + x - 3) - (x - 1) \][/tex]
Distribute the negative sign through the terms in [tex]\(g(x)\)[/tex]:
[tex]\[ (F-g)(x) = 2x^2 + x - 3 - x + 1 \][/tex]
Combine like terms:
[tex]\[ (F-g)(x) = 2x^2 + (x - x) + (-3 + 1) \][/tex]
Notice that [tex]\(x - x = 0\)[/tex]:
[tex]\[ (F-g)(x) = 2x^2 + 0 - 2 \][/tex]
Simplify the expression:
[tex]\[ (F-g)(x) = 2x^2 - 2 \][/tex]
So, the answer is:
[tex]\[ (F-g)(x) = 2x^2 - 2 \][/tex]
