Question

find the difference of functions s and r shown below.\\( r(x) = -x^2 + 3x \\) \\( s(x) = 2x + 1 \\) \\( (s - r)(x) = \\) dropdown options: \\( -x^2 + 3x - 2x + 1 \\), \\( (-x^2 + 3x) - (2x + 1) \\), \\( (2x + 1) - (-x^2 + 3x) \\), \\( 2x + 1 - x^2 + 3x \\)

Explanation:

Step1: Recall the formula for the difference of two functions

The difference of two functions \((s - r)(x)\) is defined as \(s(x)-r(x)\).
Given \(s(x) = 2x + 1\) and \(r(x)=-x^{2}+3x\), we substitute these into the formula:
\((s - r)(x)=s(x)-r(x)=(2x + 1)-(-x^{2}+3x)\)

Step2: Analyze the other options

• Option 1: \(-x^{2}+3x - 2x + 1\) is \(r(x)-2x + 1\), not \((s - r)(x)\).
• Option 2: \((-x^{2}+3x)-(2x + 1)\) is \(r(x)-s(x)\), which is \((r - s)(x)\), not \((s - r)(x)\).
• Option 4: \(2x + 1 - x^{2}+3x\) is incorrect as it does not follow the subtraction of \(r(x)\) properly.

Answer:

\((2x + 1)-(-x^{2}+3x)\) (the third option in the dropdown)