To find [tex]\((f - g)(x)\)[/tex], we must first understand what [tex]\((f - g)(x)\)[/tex] means. It means we need to subtract the function [tex]\(g(x)\)[/tex] from the function [tex]\(f(x)\)[/tex].
The given functions are:
[tex]\[ f(x) = 2x^2 + x - 3 \][/tex]
[tex]\[ g(x) = x - 1 \][/tex]
Now let's evaluate [tex]\((f - g)(x)\)[/tex]:
1. Write down the expression for [tex]\((f - g)(x)\)[/tex]:
[tex]\[ (f - g)(x) = f(x) - g(x) \][/tex]
2. Substitute [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] into this expression:
[tex]\[ (f - g)(x) = (2x^2 + x - 3) - (x - 1) \][/tex]
3. Distribute the negative sign across [tex]\(g(x)\)[/tex]:
[tex]\[ (f - g)(x) = 2x^2 + x - 3 - x + 1 \][/tex]
4. Combine like terms (the terms with [tex]\(x\)[/tex] and constants):
[tex]\[ (f - g)(x) = 2x^2 + (x - x) + (-3 + 1) \][/tex]
[tex]\[ (f - g)(x) = 2x^2 + 0x - 2 \][/tex]
[tex]\[ (f - g)(x) = 2x^2 - 2 \][/tex]
Therefore, the function [tex]\((f - g)(x)\)[/tex] is:
[tex]\[ (f - g)(x) = 2x^2 - 2 \][/tex]
Now, to find the value of [tex]\((f - g)(x)\)[/tex] at [tex]\(x = 1\)[/tex]:
[tex]\[ (f - g)(1) = 2(1)^2 - 2 \][/tex]
[tex]\[ (f - g)(1) = 2 \cdot 1 - 2 \][/tex]
[tex]\[ (f - g)(1) = 2 - 2 \][/tex]
[tex]\[ (f - g)(1) = 0 \][/tex]
Thus, the value of [tex]\((f - g)(x)\)[/tex] at [tex]\(x = 1\)[/tex] is:
[tex]\[ (f - g)(1) = 0 \][/tex]
Therefore, the numerical result is:
[tex]\[ 0 \][/tex]
