Question

for what interval is the value of \\((f - g)(x)\\) negative?

\\((-\infty, -1)\\)
\\((-\infty, 2)\\)
\\((0, 3)\\)
\\((2, \infty)\\)

Explanation:

Define the difference function

Using the Function Subtraction knowledge point

$$ (f - g)(x) = f(x) - g(x) $$

$$ (f - g)(x) = (x - 3) - (-0.5x) = 1.5x - 3 $$

Set up the inequality

To find where the value is negative:

$$ 1.5x - 3 < 0 $$

Solve for x

$$ 1.5x < 3 $$

$$ x < 2 $$

Express in interval notation

$$ x \in (-\infty, 2) $$

Answer:

• (A) \((-\infty, -1)\)
• (B) \((-\infty, 2)\) (Correct answer)
• (C) \((0, 3)\)
• (D) \((2, \infty)\)