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Chapter 4: Problem 128

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

Short Answer

Expert verified
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 by step solution

01

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 \(0 < x< 1\).
02

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 \(-3 < x < -2\).
03

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 \(1,0\).

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