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.
a. the vertical asymptote shifts 2 units to the left.
b. ( g(x) ) is translated 2 units to the left.
c. the vertical asymptote shifts 2 units to the right.
d. ( g(x) ) is translated 1 unit to the left.
e. ( g(x) ) is translated 1 unit downward.
f. ( g(x) ) is translated 2 units to the right.
g. ( g(x) ) is translated 2 units upward.

Explanation:

Step1: Analyze horizontal translation

For a function \(y = f(x + c)\) (\(c>0\)), the graph of \(y = f(x)\) is translated \(c\) units to the left.
For \(h(x)=\ln(x + 2)\), compared with \(g(x)=\ln x\), it is translated \(2\) units to the left.
The vertical asymptote of \(y = \ln x\) is \(x = 0\), and for \(y=\ln(x + 2)\), the vertical asymptote is \(x=-2\) (shifted \(2\) units to the left).

Step2: Analyze vertical translation

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

Answer:

A. The vertical asymptote shifts 2 units to the left.
B. \(g(x)\) is translated 2 units to the left.
E. \(g(x)\) is translated 1 unit downward.