Let's find the difference of the functions [tex]\( s(x) \)[/tex] and [tex]\( r(x) \)[/tex].
Given:
[tex]\[ r(x) = -x^2 + 3x \][/tex]
[tex]\[ s(x) = 2x + 1 \][/tex]
We need to find the expression [tex]\((s - r)(x)\)[/tex], which is the difference of [tex]\( s(x) \)[/tex] and [tex]\( r(x) \)[/tex].
To do this, we subtract [tex]\( r(x) \)[/tex] from [tex]\( s(x) \)[/tex]:
[tex]\[
(s - r)(x) = s(x) - r(x)
\][/tex]
Substitute the given expressions for [tex]\( s(x) \)[/tex] and [tex]\( r(x) \)[/tex] into the equation:
[tex]\[
(s - r)(x) = (2x + 1) - (-x^2 + 3x)
\][/tex]
Now, distribute the negative sign across the second term:
[tex]\[
(s - r)(x) = 2x + 1 + x^2 - 3x
\][/tex]
Combine like terms:
[tex]\[
(s - r)(x) = x^2 + 2x - 3x + 1
\][/tex]
[tex]\[
(s - r)(x) = x^2 - x + 1
\][/tex]
Thus, the difference of the functions [tex]\( s(x) \)[/tex] and [tex]\( r(x) \)[/tex] is:
[tex]\[
(s - r)(x) = x^2 - x + 1
\][/tex]
