To find [tex]\((f + g)(x)\)[/tex], we need to combine the functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex].
Given:
[tex]\[ f(x) = \frac{x}{2} - 2 \][/tex]
[tex]\[ g(x) = 2x^2 + x - 3 \][/tex]
Our goal is to add these two expressions together, that is,
[tex]\[ (f + g)(x) = f(x) + g(x) \][/tex]
Let's write out the functions with the same variable [tex]\(x\)[/tex]:
[tex]\[ (f + g)(x) = \left( \frac{x}{2} - 2 \right) + \left( 2x^2 + x - 3 \right) \][/tex]
To add these expressions, we combine like terms:
1. Combine the [tex]\(x^2\)[/tex] terms:
[tex]\[ 2x^2 \][/tex]
2. Combine the [tex]\(x\)[/tex] terms:
[tex]\[ \frac{x}{2} + x = \frac{x}{2} + \frac{2x}{2} = \frac{3x}{2} \][/tex]
3. Combine the constant terms:
[tex]\[ -2 - 3 = -5 \][/tex]
Putting these together, we get:
[tex]\[ 2x^2 + \frac{3x}{2} - 5 \][/tex]
So, the resulting function [tex]\((f + g)(x)\)[/tex] is:
[tex]\[ (f + g)(x) = 2x^2 + \frac{3x}{2} - 5 \][/tex]
This is our final expression.
