Problem 103 - 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 graph of \(g(x)=\ln x + 3\) is a vertical shift up by 3 units of the graph of \(f(x)=\ln x\).

Step 1: Understanding the Logarithmic Function

The function \(f(x)=\ln x\) is the natural logarithm function. This function is defined for \(x > 0\) and the graph crosses the y-axis at \(x = 1\).

Step 2: Plotting f(x)

Using the information from Step 1, plot \(f(x)=\ln x\) on a graph. The graph of \(f(x)\) passes through the point (1,0) and is increasing as \(x\) increases. The graph will descend towards the negative y-axis but will never touch it.

Step 3: Understanding the Transformation

The function \(g(x)=\ln x + 3\) is a vertical translation of the function \(f(x)\). This means that the graph of \(g(x)\) is the same as the graph of \(f(x)\), but moved up 3 units on the y-axis.

Step 4: Plotting g(x)

The graph of \(g(x)\) will pass through the point (1,3) and will proceed the same way as the graph of \(f(x)\), but 3 units above it. It descends towards the horizontal line \(y = 3\) as \(x\) approaches 0.

Step 5: Describing the Relationship

The graph of \(g(x)\) is a vertical shift up of the graph of \(f(x)\) by 3 units. That is, every point on the graph of \(f(x)\) is moved 3 units up to get the corresponding point on the graph of \(g(x)\).