Problem 102 - Introductory and Intermediate Algebra for College Students 4th

Question

Graph \(f\) and \(g\) in the same viewing rectangle. Then describe the relationship of the graph of g to the graph of \(f\). $$f(x)=\ln x, g(x)=\ln (x+3)$$

Step-by-Step Solution

Answer

The function \(g(x)=\ln (x+3)\) is a horizontal shift of the function \(f(x)=\ln x\), three units to the left. They share the same shape but have different locations on the graph. \(f(x)\) is defined for \(x > 0\), crossing the x-axis at \(x = 1\), while \(g(x)\) is defined for \(x > -3\), crossing the x-axis at \(x = -2\). Both graphs are increasing and concave down.

Step 1: Graphing Function f

First, graph \(f(x)=\ln x\). The natural logarithm function, \(f(x)=\ln x\), is only defined where \(x\) is greater than zero. It crosses the x-axis at \(x=1\). The graph is increasing and concave down.

Step 2: Graphing Function g

Now, graph \(g(x)=\ln (x+3)\). The function \(g(x)=\ln (x+3)\) is obtained by shifting the graph of \(f(x)=\ln x\) three units to the left. So, it is defined where \(x > -3\). It also crosses the x-axis at \(x=-2\).

Step 3: Comparing the Graphs of f and g

Looking at both graphs, it is noticeable that the graph of \(g(x)\) is a transformation of the graph of \(f(x)\). Particularly, it is a horizontal shift of the graph of \(f(x)\) 3 units to the left. Therefore, the graph of \(g(x)\) and the graph of \(f(x)\) have the same shape, but the graph of \(g(x)\) is located 3 units to the left of the graph of \(f(x)\).