To find the difference between the functions [tex]\( s(x) \)[/tex] and [tex]\( r(x) \)[/tex], we need to subtract [tex]\( r(x) \)[/tex] from [tex]\( s(x) \)[/tex].
Given the functions:
[tex]\[ r(x) = -x^2 + 3x \][/tex]
[tex]\[ s(x) = 2x + 1 \][/tex]
We want to determine [tex]\( (s - r)(x) \)[/tex], which is calculated as follows:
[tex]\[ (s - r)(x) = s(x) - r(x) \][/tex]
Substituting the given functions into this expression:
[tex]\[ (s - r)(x) = (2x + 1) - (-x^2 + 3x) \][/tex]
To simplify, distribute the minus sign through the parentheses:
[tex]\[ (s - r)(x) = 2x + 1 + x^2 - 3x \][/tex]
Next, combine like terms:
[tex]\[ (s - r)(x) = x^2 + (2x - 3x) + 1 \][/tex]
[tex]\[ (s - r)(x) = x^2 - x + 1 \][/tex]
Therefore, the difference of the functions [tex]\( s \)[/tex] and [tex]\( r \)[/tex] is:
[tex]\[ (s - r)(x) = x^2 - x + 1 \][/tex]
This is the resulting expression for [tex]\( (s - r)(x) \)[/tex].
