Question

question 9 of 10
below is the graph of ( f(x)=ln (x) ). how would you
describe the graph of ( g(x)=\frac{1}{3} ln (x) )?

a. ( g(x) ) compresses ( f(x) ) by a factor of ( \frac{1}{3} ).
b. ( g(x) ) shifts ( f(x) ) to the left ( \frac{1}{3} ) units.
c. ( g(x) ) shifts ( f(x) ) vertically ( \frac{1}{3} ) units.
d. ( g(x) ) stretches ( f(x) ) vertically by a factor of ( \frac{1}{3} ).

Explanation:

Step1: Recall vertical transformation rules

For a function \(y = a\cdot f(x)\), if \(0 < a< 1\), the graph of \(y = f(x)\) is vertically compressed by a factor of \(a\). If \(a>1\), the graph of \(y = f(x)\) is vertically stretched by a factor of \(a\).

Step2: Analyze \(g(x)=\frac{1}{3}\ln(x)\) compared to \(f(x)=\ln(x)\)

Here \(a = \frac{1}{3}\) and \(0<\frac{1}{3}<1\). So the graph of \(g(x)\) is a vertical compression of the graph of \(f(x)\) by a factor of \(\frac{1}{3}\).

Answer:

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