Problem 97 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the graph of \(f(x)=\ln x\) to describe the transformation that yields the graph of \(g\). $$g(x)=\ln (x-1)+2$$

Step-by-Step Solution

Answer

The transformation of the function \(f(x)=\ln x\) to form the function \(g(x)=\ln (x-1)+2\) involves a horizontal shift to the right by 1 unit and a vertical shift upwards by 2 units.

Step 1: Identify the function

Identify the original function. Here, the original function is \(f(x)=\ln x\), which is a logarithm with base \(e\). The function \(g(x)=\ln (x-1)+2\) would be obtained by transforming \(f(x)\).

Step 2: Identify the transformation

Identify the type of transformation performed on the original function \(f(x)\) to get \(g(x)\). The functions are related by \(g(x)=\ln (x-1)+2\), this shows a two-part transformation: a horizontal shift to the right by 1 unit, shown by \((x-1)\) and a vertical shift upwards by 2 units, as shown by \(+2\) in the function \(g(x)\).

Step 3: Graph the transformed function

To get the graph of \(g(x)\), first plot the graph of the original function and then perform the transformation on the graph. Move each point on the original function to the right by 1 unit and upwards by 2 units. This will give you the graph of \(g(x)\).

Key Concepts

Logarithmic Function

The logarithmic function is a fascinating mathematical concept often depicted in graphs as a curve. A common example is the natural logarithm, denoted as \( f(x) = \ln x \), which uses the base \( e \), an irrational constant approximately equal to 2.71828.
Logarithmic functions are the inverse of exponential functions. This means that if you have an exponential function \( y = a^x \), its inverse is a logarithmic function \( x = \log_a y \).
Key features of the logarithmic graph include:

• It passes through the point (1,0), because \( \ln 1 = 0 \).
• The domain is \( x > 0 \), meaning it is only defined for positive \( x \) values.
• The range is all real numbers, so the graph keeps rising but never crosses the x-axis, getting closer and closer to it as \( x \) decreases towards zero.

Understanding these features will help you analyze various transformations applied to the logarithmic function.

Horizontal Shift

A horizontal shift involves moving the entire graph of a function left or right. This type of transformation is evident in functions of the form \( g(x) = f(x - h) \), where \( h \) is the number of units you shift the graph.
In our example, consider \( g(x) = \ln(x-1) \). The \((x-1)\) indicates a horizontal shift to the right by 1 unit.

• If \( h > 0 \), the graph shifts right.
• If \( h < 0 \), the graph shifts left.

This shift alters the domain of your function. For \( \ln(x-1) \), the domain changes from \( x > 0 \) to \( x > 1 \). This process helps create a new starting point for the log function's iconic shape while maintaining its general behavior.

Vertical Shift

A vertical shift occurs when you move a graph up or down without changing its form. In the case of our exercise, the transformation that results in the graph \( g(x) = \ln(x-1) + 2 \) includes such a shift. The "+2" signifies that every point along the graph is to be raised by 2 units.

• Addition results in an upward shift.
• Subtraction results in a downward shift.

This modification affects the range but does not impact the domain or horizontal position of the function.
For \( g(x) = \ln(x-1) + 2 \), the range changes from all real numbers to all real numbers plus 2. This means that each value of \( g(x) \) is simply the corresponding value of \( \ln(x-1) \) moved up by 2 steps on the y-axis. Understanding vertical shifts is essential for accurately plotting and interpreting transformed graphs.