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]. Let's define the functions:
[tex]\[ r(x) = -x^2 + 3x \][/tex]
[tex]\[ s(x) = 2x + 1 \][/tex]
We aim to find [tex]\((s - r)(x)\)[/tex]:
[tex]\[
(s - r)(x) = s(x) - r(x)
\][/tex]
Substitute [tex]\(r(x)\)[/tex] and [tex]\(s(x)\)[/tex] into the equation:
[tex]\[
(s - r)(x) = (2x + 1) - (-x^2 + 3x)
\][/tex]
Distribute the negative sign through the [tex]\(r(x)\)[/tex] term:
[tex]\[
(s - r)(x) = 2x + 1 + x^2 - 3x
\][/tex]
Now, combine like terms:
[tex]\[
(s - r)(x) = x^2 + (2x - 3x) + 1
\][/tex]
Simplify the expression inside the parentheses:
[tex]\[
(s - r)(x) = x^2 - x + 1
\][/tex]
Hence, the difference [tex]\((s - r)(x)\)[/tex] is:
[tex]\[
(s - r)(x) = x^2 - x + 1
\][/tex]
This is the simplified form of the resulting function when subtracting [tex]\(r(x)\)[/tex] from [tex]\(s(x)\)[/tex].
