Problem 128 - 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 graph of function \(g(x)\) is a horizontal shift of the graph of function \(f(x)\) 3 units to the left. So, the graph of \(g(x)\) is the transformation of the graph of \(f(x)\).

Step 1: Graph of Function \(f(x)\)

To sketch the graph of \(f(x) = \ln x\), remember that the natural log function has a vertical asymptote at \(x=0\), and it passes through the point \(1,0\). Keep in mind that the curve increases slowly for \(x>1\) and it decreases slowly for \(0 < x< 1\).

Step 2: Graph of Function \(g(x)\)

The function \(g(x) = \ln(x+3)\) is a horizontal shift of the function \(f(x) = \ln x\). This is because adding a constant within the logarithmic function results in a horizontal shift. Specifically, \(g(x) = \ln(x+3)\) is a shift of 3 units to the left from \(f(x) = \ln x\). Thus, you can sketch \(g(x)\) by taking the graph of \(f(x)\) and shifting it 3 units to the left. This means that the vertical asymptote will be at \(x=-3\) and the graph will pass through the point \(-2,0\). The graph also increases slowly for \(x > -2\) and decreases slowly for \(-3 < x < -2\).

Step 3: Describe the Relationship

Comparing the graphs, it is clear that the graph of \(g(x)\) is a transformation of the graph of \(f(x)\). More specifically, \(g(x) = \ln (x + 3)\) is a horizontal shift of 3 units to the left of \(f(x)\). This is confirmed by the vertical asymptote shifting from \(x=0\) to \(x=-3\) and the graph passing through \(-2,0\) instead of \(1,0\).