Problem 129 - College Algebra Essentials

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 the graph of the function \(f(x) = \ln x\) shifted upwards by three units.

Step 1: Graphing the function \(f(x) = \ln x\)

Begin by plotting the graph of the function \(f(x) = \ln x\). The logarithm of \(x\) is undefined for \(x \leq 0\). The graph of \(f(x) = \ln x\) has an x-intercept at (1,0). This is because the natural logarithm of 1 is 0 (since \(e^0 = 1\)). As \(x\) approaches 0, \(f(x)\) approaches negative infinity.

Step 2: Graphing the function \(g(x) = \ln x + 3\)

Next is to graph the function \(g(x) = \ln x + 3\). The function \(g(x) = \ln x + 3\) is a vertical shift of the function \(f(x) = \ln x\) by 3 units upward because the '+3' indicates that every y-value on the original graph is increased by 3. Thus, in the graph, this will be observed as a shift of the graph of \(f(x)\) 3 units up.

Step 3: Describing the relationship of the graph of \(g\) to the graph of \(f\)

To describe the relationship between the two graphs, one must observe how the graph of \(f(x)\) has transformed to give the graph of \(g(x)\). In this case, adding a 3 to the original function \(f(x) = \ln x\) results in the graph \(g(x) = \ln x + 3\), which is the upward shift of \(f(x)\) by 3 units.