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Chapter 12: Problem 103

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

Step by step solution

01

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 \(x = 1\).
02

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.
03

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.
04

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.
05

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 \(g(x)\).

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Most popular questions from this chapter

Graph each of the following functions in the same viewing rectangle and then place the functions in order from the one that increases most slowly to the one that increases most rapidly. $$y=x, y=\sqrt{x}, y=e^{x}, y=\ln x, y=x^{x}, y=x^{2}$$

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