Problem 129 Graph $f$ and $g$ in the sam... [FREE SOLUTION]

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College Algebra

Robert Blitzer

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 4: Problem 129

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 $g(x) = \ln x + 3$ is identical to the graph of function $f(x) = \ln x$ but shifted upwards by 3 units.

Step by step solution

Graphing the function $f(x) = \ln x$

To graph this function, select a range of x values (preferably both positive and negative) and calculate the corresponding y values using the $\ln x$ function. Then plot these values on a graph, using the x values as the horizontal coordinates and the y values as the vertical coordinates.

Graphing the function $g(x) = \ln x + 3$

Repeat the same process as in Step 1 but this time use the function $\ln x + 3$. This will result in values higher by 3 for the same x coordinates compared to the first function. Again plot these values on the same graph.

Observation and analysis of the two functions

Compare the two plots. The graph of $g(x) = \ln x + 3$ will be identical to the graph of $f(x) = \ln x$ but shifted upwards by 3 units. This is the effect of the +3 in the equation of $g(x)$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Natural Logarithm

Understanding the natural logarithm, represented as $ \ln x $, is essential in various areas of mathematics, especially in calculus and algebra. The natural logarithm is the inverse function of the exponential function $ e^x $. This means that if $ y = \ln x $, then $ e^y = x $.

The function $ \ln x $ is defined only for positive values of $ x $. This is because logarithms of zero or negative numbers are undefined in the realm of real numbers.
The natural logarithm of 1 is always 0, i.e., $ \ln 1 = 0 $. This is because $ e^0 = 1 $.
As $ x $ approaches infinity, $ \ln x $ increases without bound, though at a decreasing rate.

Practically, the curve of the natural logarithm is gently increasing, never quite reaching zero but always trending upward, creating a smooth curve that flattens as $ x $ increases.

Vertical Shift

A vertical shift in functions is a simple yet important transformation. It involves moving a graph up or down along the y-axis without altering its shape.

Addition of a constant $ c $ to a function, as in $ g(x) = \ln x + c $, shifts the graph of the function upward by $ c $ units.
Similarly, subtracting $ c $ from the function shifts the graph downward by $ c $ units.

In the given exercise, you have the function $ f(x) = \ln x $ and $ g(x) = \ln x + 3 $. Here, the graph of $ g(x) $ is simply the graph of $ f(x) $ shifted 3 units upward. This transformation is visually represented by parallel movements of the graph along the vertical axis, known as translation.

Function Graphing

Graphing functions is a crucial skill for visualizing and understanding their behavior. When graphing functions like $ \ln x $ and $ \ln x + 3 $, it's important to follow a systematic approach:

Select a range of $ x $ values, keeping in mind that for $ \ln x $, you must choose positive values only.
Calculate corresponding $ y $ values using each function's formula.
Plot these $ (x, y) $ points on a coordinate plane.

For $ f(x) = \ln x $, start by plotting points like $ (1, 0) $, $ (2, \ln 2) $, and so on, to gradually build the curve. Then, graph $ g(x) = \ln x + 3 $, which involves taking each point from $ f(x) $ and moving it up 3 units, reflecting the vertical shift. Overlapping these graphs allows you to compare them directly and observe how transformations affect function behavior visually.

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91影视

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

Step by step solution

Graphing the function \(f(x) = \ln x\)

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

Observation and analysis of the two functions

Key Concepts

Natural Logarithm

Vertical Shift

Function Graphing

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