Sure, I can help you find the difference between the functions [tex]\( s(x) \)[/tex] and [tex]\( r(x) \)[/tex]. The functions are given as follows:
[tex]\[ r(x) = -x^2 + 3x \][/tex]
[tex]\[ s(x) = 2x + 1 \][/tex]
We need to find [tex]\( (s - r)(x) \)[/tex], which means we are looking for the function [tex]\( s(x) \)[/tex] minus the function [tex]\( r(x) \)[/tex]. Let's perform the subtraction step-by-step.
1. Start by writing the expression for [tex]\( (s - r)(x) \)[/tex]:
[tex]\[ (s - r)(x) = s(x) - r(x) \][/tex]
2. Substitute the given expressions for [tex]\( s(x) \)[/tex] and [tex]\( r(x) \)[/tex]:
[tex]\[ (s - r)(x) = (2x + 1) - (-x^2 + 3x) \][/tex]
3. Distribute the negative sign across the terms in [tex]\( r(x) \)[/tex]:
[tex]\[ (s - r)(x) = 2x + 1 + x^2 - 3x \][/tex]
4. Combine like terms to simplify the expression:
[tex]\[ (s - r)(x) = x^2 + 2x - 3x + 1 \][/tex]
5. Combine the [tex]\( x \)[/tex] terms:
[tex]\[ (s - r)(x) = x^2 - x + 1 \][/tex]
Therefore, the difference between the functions [tex]\( s(x) \)[/tex] and [tex]\( r(x) \)[/tex] is:
[tex]\[ (s - r)(x) = x^2 - x + 1 \][/tex]
This is the simplified form of the difference of the given functions.
