Question

Sketch the graph of the function and state its domain. $$ f(x)=2+\ln x $$

Short answer

The graph of \(f(x)=2+\ln x\) is the graph of \(y=\ln x\), shifted upward by 2 units. The domain is \(x>0\).

Step-by-step solution

Understand the Function

First, understand that \(f(x)=2+\ln x\) is a logarithmic function. The base of the logarithm is the mathematical constant e, approximately equal to 2.71828. Logarithmic functions are the inverses of exponential functions, and the general form for a natural logarithmic function is \(f(x) = \ln x\). When a constant is added (in this case 2), it results in a vertical shift of the graph. The positive constant shifts the graph upward.

Identify the Domain

The natural logarithm, \(\ln x\), is only defined for positive real numbers. Therefore, \(f(x)=2+\ln x\) will only be defined for \(x > 0\). This is the domain of the function.

Sketch the Graph

Start by sketching \(y=\ln x\), which is a curve that gets arbitrarily close to the y-axis but never touches or crosses it (the y-axis is a vertical asymptote), and which passes through the point (1,0). Each point on \(y=\ln x\) is shifted upward by 2 units to create the graph of \(f(x)=2+\ln x\). Also, mark the vertical asymptote x=0, which is a vertical line through x=0, where the function is undefined.