Sure, let's find the function [tex]\( f(x) - g(x) \)[/tex] step by step.
Given:
[tex]\[ f(x) = x^3 - 2x^2 + 3x - 5 \][/tex]
[tex]\[ g(x) = x^2 + x - 1 \][/tex]
We are required to determine [tex]\( f(x) - g(x) \)[/tex].
Step 1: Write down the expressions for [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex].
[tex]\[ f(x) = x^3 - 2x^2 + 3x - 5 \][/tex]
[tex]\[ g(x) = x^2 + x - 1 \][/tex]
Step 2: Subtract [tex]\( g(x) \)[/tex] from [tex]\( f(x) \)[/tex].
[tex]\[ f(x) - g(x) = (x^3 - 2x^2 + 3x - 5) - (x^2 + x - 1) \][/tex]
Step 3: Distribute the negative sign across the terms inside the parenthesis containing [tex]\( g(x) \)[/tex].
[tex]\[ f(x) - g(x) = x^3 - 2x^2 + 3x - 5 - x^2 - x + 1 \][/tex]
Step 4: Combine like terms.
[tex]\[ = x^3 - (2x^2 + x^2) + (3x - x) - (5 - (-1)) \][/tex]
[tex]\[ = x^3 - 3x^2 + 2x - 4 \][/tex]
So, the result for [tex]\( f(x) - g(x) \)[/tex] is:
[tex]\[ f(x) - g(x) = x^3 - 3x^2 + 2x - 4 \][/tex]
