Problem 129 - College Algebra

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

Step 1: 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.

Step 2: 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.

Step 3: 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 \(g(x)\).

Key Concepts

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.