Problem 66 - College Algebra

Question

The figure shows the graph of \(f(x)=\ln x\). 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. (GRAPH CANNOT COPY). $$ g(x)=\ln (x+1) $$

Step-by-Step Solution

Answer

The function \(g(x) = \ln(x+1)\) is a shift to the left by 1 unit of the function \(f(x) = \ln(x)\). The equation of the vertical asymptote is \(x = -1\). The domain of \(g(x)\) is \((-1, +\infty)\), and its range is \((-\infty, +\infty)\).

Step 1: Understand the transformation

The function \(g(x) = \ln(x+1)\) is a transformation of the function \(f(x) = \ln(x)\). This is a shift to the left by one unit because of the \(+1\) within the logarithm. This does not affect the shape of the graph, but moves its position.

Step 2: Graph the function

To draw the graph of \(g(x)\), take the graph of \(f(x)\) (which increases slowly, going through the point (1,0), and has a vertical asymptote at x=0), and shift it left by 1 unit. This means the graph of \(g(x)\) also increases slowly, goes through the point (0,0), and has a vertical asymptote at x=-1. The graph cannot be copied in text format.

Step 3: Write the equation of the asymptotes

The vertical asymptote is a vertical line that the graph approaches as \(x\) gets very small. The graph never crosses this line. For this function, the vertical asymptote is at \(x=-1\), so the equation of the asymptote is \(x=-1\).

Step 4: Determine the domain and range

The domain of a function is the set of all allowed \(x\)-values, and the range is the set of all resulting \(y\)-values. In this case, \(x\) must be greater than -1 (because the logarithm of a non-positive number is undefined), so the domain is \((-1, +\infty)\). The function can output any real number, so the range is \((-\infty, +\infty)\).

Key Concepts

Domain and Range

Understanding the domain and range of a function is crucial as it tells us which values are allowed as input (domain) and which values can be the outcome of the function (range).
For the function \( g(x) = \ln(x+1) \), the domain is determined by the requirement that the argument of the logarithm must be positive. This condition means:

• \( x+1 > 0 \)
• \( x > -1 \)

Thus, the domain of \( g(x) \) is all real numbers greater than -1, written as \((-1, \infty)\).
For the range, logarithmic functions can output any real number, meaning the set of \(y\)-values is unrestricted. Hence, the range of \( g(x) \) is \(( -\infty, +\infty )\).
These aspects help us understand where the function's graph will lie in the coordinate plane, and which parts of the plane it can cover.

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions. They help us solve equations that involve exponential growth or decay.
The basic form of a logarithmic function is \( f(x) = \ln(x) \), where \( \ln \) signifies the natural logarithm, which has base \( e \), an irrational constant approximately equal to 2.718.
Key properties of logarithmic functions include:

• As \(x\) approaches zero from the positive side, the value of \(\ln(x)\) decreases towards negative infinity.
• For \(x > 1\), \(\ln(x)\) results in positive values. At \( x = 1 \), \( \ln(1) = 0 \).

Transformations like shifting, compressing or stretching can alter the appearance of the graph but do not change these fundamental properties.
This makes logarithms very useful for modeling various real-world phenomena, such as pH in chemistry or the Richter scale for measuring earthquakes.

Vertical Asymptotes

Vertical asymptotes are important features of certain functions, including logarithmic functions. They are vertical lines that the curve approaches but never actually meets.
For the function \(g(x) = \ln(x+1)\), a vertical asymptote appears at \(x = -1\). This is because adding 1 to \(x\) changes the point where the logarithmic function becomes undefined, moving the asymptote of \(\ln(x)\) from \(x = 0\) to \(x = -1\).
Vertical asymptotes give insights into the "forbidden" values of \(x\) that are not included in the domain. When sketching the graph, they help in understanding the behavior of the function as \(x\) gets very close to these specific points on the x-axis.
It's important to remember that while the graph approaches the asymptote, it does not cross or touch it, highlighting a kind of boundary region in the function's graph.