To find the difference between the functions [tex]\( s(x) \)[/tex] and [tex]\( r(x) \)[/tex], we need to compute [tex]\( (s - r)(x) \)[/tex]. Here are the detailed steps:
1. Express the functions [tex]\( r(x) \)[/tex] and [tex]\( s(x) \)[/tex]:
[tex]\[
r(x) = -x^2 + 3x
\][/tex]
[tex]\[
s(x) = 2x + 1
\][/tex]
2. Find the difference [tex]\( (s - r)(x) \)[/tex]:
[tex]\[
(s - r)(x) = s(x) - r(x)
\][/tex]
3. Substitute the expressions for [tex]\( s(x) \)[/tex] and [tex]\( r(x) \)[/tex]:
[tex]\[
(s - r)(x) = (2x + 1) - (-x^2 + 3x)
\][/tex]
4. Simplify the expression by distributing and combining like terms:
[tex]\[
(s - r)(x) = 2x + 1 + x^2 - 3x
\][/tex]
[tex]\[
(s - r)(x) = x^2 + 2x - 3x + 1
\][/tex]
[tex]\[
(s - r)(x) = x^2 - x + 1
\][/tex]
Therefore, the difference [tex]\( (s - r)(x) \)[/tex] is:
[tex]\[
(s - r)(x) = x^2 - x + 1
\][/tex]
