Question

find the difference of functions s and r shown below.
r(x) = -x² + 3x \t\ts(x) = 2x + 1
(s - r)(x) =
options:
-x² + 3x - 2x + 1
(-x² + 3x) - (2x + 1)
(2x + 1) - (-x² + 3x)
2x + 1 - x² + 3x

Explanation:

Step1: Recall function subtraction rule

The difference of two functions \((s - r)(x)\) is defined as \(s(x)-r(x)\).
Given \(r(x)=-x^{2}+3x\) and \(s(x) = 2x+1\), so \((s - r)(x)=s(x)-r(x)=(2x + 1)-(-x^{2}+3x)\)

Step2: Analyze each option

• Option 1: \(-x^{2}+3x - 2x + 1\) is \(r(x)-s(x)\) (incorrect sign handling), so wrong.
• Option 2: \((-x^{2}+3x)-(2x + 1)\) is \(r(x)-s(x)\), wrong.
• Option 3: \((2x + 1)-(-x^{2}+3x)\) matches \(s(x)-r(x)\), correct.
• Option 4: \(2x + 1-x^{2}+3x\) is \(s(x)+r(x)\) (incorrect operation), wrong.

Answer:

(2x + 1) - (-x² + 3x)