Question

what is the range of function \\(g\\) if \\(g(x) = -f(x) - 5\\)?

\\(\bigcirc \\{y | -\infty < y < \infty\\}\\)
\\(\bigcirc \\{y | -\infty < y < -5\\}\\)
\\(\bigcirc \\{y | -5 < y < \infty\\}\\)
\\(\bigcirc \\{y | -5 < y < 0\\}\\)

Explanation:

Identify the range of the original function \(f(x)\)

$$ \text{Range}(f) = (0, \infty) \quad \text{or} \quad 0 < f(x) < \infty $$

Apply the vertical reflection transformation

$$ y_1 = -f(x) \implies -\infty < y_1 < 0 $$

Apply the vertical translation transformation

$$ g(x) = -f(x) - 5 \implies -\infty < g(x) < -5 $$

Answer:

• (A) \(\{y | -\infty < y < \infty\}\)
• (B) \(\{y | -\infty < y < -5\}\) (Correct answer)
• (C) \(\{y | -5 < y < \infty\}\)
• (D) \(\{y | -5 < y < 0\}\)