Problem 66 - Algebra and Trigonometry

Question

The figure shows the graph of \(f(x)=\ln x .\) In Exercises \(65-74\) use transformations of this graph to graph each function. Graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. $$ g(x)=\ln (x+1) $$

Step-by-Step Solution

Answer

The graph of \(g(x) = \ln(x+1)\) is a shift of the graph of \(f(x) = \ln(x)\) 1 unit to the left. The vertical asymptote of the function is at \(x=-1\). The domain of the function is \(x > -1\) and the range is all real numbers.

Step 1: Identify the Transformation

The transformation from \(f(x)=\ln(x)\) to \(g(x)=\ln(x+1)\) involves a horizontal shift, specifically a shift to the left by one unit. This happens because x in the function is replaced by (x+1).

Step 2: Graphing the Function

Start with the graph of \(y=\ln(x)\), and then shift every point on this graph one unit to the left to graph \(y=\ln(x+1)\). The point (1, 0) shifts to the point (0, 0), and the y-axis serves as a vertical asymptote.

Step 3: Identify the Asymptote

An asymptote is a line that the graph approaches but never touches. In this case, with the function \(g(x) = \ln(x+1)\), the vertical asymptote is the line \(x=-1\), due to the shift of the function.

Step 4: Determine the Range and Domain

The domain of a function is the set of all possible x-values. In this case, the domain of \(g(x) = \ln(x+1)\) is \(x > -1\), because logarithms are only defined for positive numbers. The range of a function is the set of all possible y-values. For the function \(g(x)=\ln(x+1)\), as with any logarithmic function, the range is all real numbers.

Key Concepts

Horizontal Shift

A horizontal shift is a transformation that moves the graph of a function left or right. In the case of the function transformation from \(f(x) = \ln(x)\) to \(g(x) = \ln(x+1)\), the graph is shifted left by one unit. This happens because the expression \((x+1)\) inside the logarithm suggests that we start at earlier values of \(x\). Essentially, when you add a positive constant inside the expression \( (x + c) \), you shift the graph to the left by \(c\) units.

For example:

• The point \((1, 0)\) on the graph of \(\ln(x)\) moves to \((0, 0)\) on the graph of \(\ln(x+1)\).
• Similarly, the point \((e, 1)\) — where \(e\) is the base of the natural logarithm, approximately 2.718 — now appears at \((e-1, 1)\).

Understanding horizontal shifts is crucial as it allows us to predict how function graphs will behave without recalculating everything from scratch. The entire graph moves leftward in relation to this shift, while maintaining its shape.

Domain and Range

The domain of a function refers to all the possible x-values that make the function defined. For logarithmic functions like \(g(x) = \ln(x+1)\), the domain is influenced by the need for the argument inside the logarithm to be greater than zero.

Here's a breakdown:

• For \(g(x) = \ln(x+1)\), the argument \((x+1)\) must be greater than zero.
• This means \(x > -1\) to keep the function defined and real.

The range of a logarithmic function is a little different. For any logarithmic function, the y-values can stretch from negative infinity to positive infinity. This is because as \(x\) values slowly increase, so do \(y\) values, without any upper boundary. At the same time, as \(x\) approaches its lower bound, the \(y\) values tend toward negative infinity. This makes the range of \(g(x) = \ln(x+1)\) all real numbers.

Vertical Asymptote

A vertical asymptote is a line that the graph of a function approaches as the x-values get very close to a certain number but never actually reaches or intersects. In the transformed function \(g(x) = \ln(x+1)\), due to the horizontal shift, the vertical asymptote shift also occurs.

For the basic function \(f(x) = \ln(x)\), the vertical asymptote is at \(x = 0\), just because logarithms aren't defined for non-positive numbers.

• When \(f(x)\) changes to \(g(x) = \ln(x+1)\), the vertical asymptote shifts too, moving one unit to the left because of the \(+1\) inside the logarithm.
• As a result, we end up with a vertical asymptote at \(x = -1\) for the function \(g(x)\).

This asymptote visually represents the boundary of the domain. The graph of \(g(x)\) will hover near this line but will never cross it, giving us pivotal insight into function behavior as \(x\) values approach the asymptote.