Question

Sketch the graph of the function and state its domain. $$ f(x)=\ln (x-1) $$

Short answer

The graph of the function \(f(x) = \ln (x - 1)\) starts at \(x = 1\) with a vertical asymptote and increases without bound as \(x\) goes towards +infinity. The domain of the function is \(x > 1\).

Step-by-step solution

Understand The Base Function

The natural logarithm function, \(\ln (x)\), has the graph of y intercept at 0 and increasing without bound as \(x\) approaches +infinity, it is undefined for \(x ≤ 0\), hence it has a vertical asymptote at \(x = 0\). When the function is \(\ln (x - 1)\), the entire graph of \(\ln (x)\) is shifted one unit to the right. This is because the \(x - 1\) in the argument of the logarithm delays the input into the logarithm function by 1.

Sketch The Graph

The graph of the function \(f(x) = \ln (x - 1)\) can be drawn by starting with the graph of \(y = \ln (x)\), and shifting all points one unit to the right. There will be a vertical asymptote at \(x = 1\), and the function increases without bound as \(x\) goes towards +infinity.

State The Domain

The domain of the function \(f(x) = \ln (x - 1)\) can be decided by considering when the function is defined. The natural logarithm is only defined for positive numbers, which means that \(x - 1 > 0\) for the function \(f(x) = \ln (x - 1)\). Solving this inequality gives \(x > 1\). Therefore, the domain of the function is \(x > 1\).