Question

below is the graph of $f(x) = \ln(x)$. how would you describe the graph of $g(x) = \frac{1}{3}\ln(x)$? graph of f(x) = ln(x) is shown \\(\bigcirc\\) a. $g(x)$ stretches $f(x)$ vertically by a factor of $\frac{1}{3}$. \\(\bigcirc\\) b. $g(x)$ compresses $f(x)$ by a factor of $\frac{1}{3}$. \\(\bigcirc\\) c. $g(x)$ shifts $f(x)$ vertically $\frac{1}{3}$ units. \\(\bigcirc\\) d. $g(x)$ shifts $f(x)$ to the left $\frac{1}{3}$ units.

Explanation:

Step1: Recall Vertical Transformation Rules

For a function \( y = a \cdot f(x) \), if \( |a| < 1 \), it's a vertical compression; if \( |a| > 1 \), it's a vertical stretch. Here, \( g(x)=\frac{1}{3}\ln(x) \) means \( a = \frac{1}{3} \), so we check the transformation.

Step2: Analyze Each Option

• Option A: Vertical stretch is when \( |a| > 1 \), but \( \frac{1}{3}<1 \), so not a stretch. Eliminate A.
• Option B: Since \( a=\frac{1}{3} \) ( \( |a| < 1 \) ), \( g(x) \) compresses \( f(x) \) vertically by factor \( \frac{1}{3} \). This matches the rule.
• Option C: Vertical shift is \( y = f(x)+k \), but here it's a scalar multiple, not a shift. Eliminate C.
• Option D: Horizontal shift is \( y = f(x - h) \), but \( g(x) \) is not a horizontal shift. Eliminate D.

Answer:

B. \( g(x) \) compresses \( f(x) \) by a factor of \( \frac{1}{3} \)