Question

identify the graph of ( y = ln x + 1 ).

Explanation:

Step1: Analyze the parent function

The parent function of \(y = \ln x+1\) is \(y=\ln x\). The domain of \(y = \ln x\) is \(x>0\), and it passes through the point \((1,0)\) since \(\ln(1)=0\).

Step2: Analyze the transformation

For the function \(y=\ln x + 1\), it is a vertical - shift of the function \(y = \ln x\). The rule for a vertical shift is \(y=f(x)+k\), where if \(k>0\), the graph of \(y = f(x)\) is shifted up by \(k\) units. For \(y=\ln x+1\), the graph of \(y = \ln x\) (which passes through \((1,0)\)) is shifted up by \(1\) unit. So the function \(y=\ln x + 1\) passes through the point \((1,1)\) (because when \(x = 1\), \(y=\ln(1)+1=0 + 1=1\)).

Answer:

The graph that passes through the point \((1,1)\) is the correct graph of \(y=\ln x+1\).