Question

the logarithmic function ( g(x)=ln x ) is transformed to ( h(x)=ln (x + 2)-1 ).
which of the following are true? select all that apply.
( square ) ( g(x) ) is translated 2 units upward
( square ) ( g(x) ) is translated 2 units to the right
( square ) ( g(x) ) is translated 2 units to the left
( square ) ( g(x) ) is translated 1 unit downward
( square ) ( g(x) ) is translated 1 unit to the left
( square ) the vertical asymptote shifts 2 units to the left.
( square ) the vertical asymptote shifts to units to the right.

Explanation:

Step1: Analyze horizontal translation

For a function \(y = f(x + c)\), if \(c>0\), the graph of \(y = f(x)\) is translated \(c\) units to the left.
For \(h(x)=\ln(x + 2)-1\) compared to \(g(x)=\ln x\), since \(c = 2>0\), \(g(x)\) is translated \(2\) units to the left.

Step2: Analyze vertical translation

For a function \(y=f(x)-d\), if \(d>0\), the graph of \(y = f(x)\) is translated \(d\) units downward.
For \(h(x)=\ln(x + 2)-1\) compared to \(g(x)=\ln x\), since \(d = 1>0\), \(g(x)\) is translated \(1\) unit downward.

Step3: Analyze vertical asymptote

The vertical asymptote of \(y=\ln x\) is \(x = 0\).
For \(y=\ln(x + 2)\), set \(x+2=0\), we get \(x=-2\). So the vertical asymptote shifts \(2\) units to the left.

Answer:

C. \(g(x)\) is translated \(2\) units to the left; D. \(g(x)\) is translated \(1\) unit downward; F. The vertical asymptote shifts \(2\) units to the left.