Problem 6 - Elementary and Intermediate Algebra

Question

a. Use a calculator to complete the table of values in the next column for \(f(x)=\ln x .\) Round to the nearest hundredth. b. Graph \(f(x)=\) In \(x .\) Note that the units on the \(x\) - and \(y\) -axes are different. c. What are the domain and range of the function? d. What is the \(x\) -intercept of the graph? What is the \(y\) -intercept? e. What is an asymptote of the graph? f. Is \(f\) increasing or decreasing? g. Is the function one-to-one? (GRAPH CANNOT COPY)

Step-by-Step Solution

Answer

The domain is \((0, \infty)\), range is \((-\infty, \infty)\); \( x \)-intercept at \( x=1 \); vertical asymptote at \( x=0 \). The function is increasing and one-to-one.

Step 1: Calculate values for the table

Use a calculator to find the values of \( f(x) = \ln x \) for a series of values of \( x \). For instance: Calculate \( \ln 1 \), \( \ln 2 \), \( \ln 3 \), \( \ln 4 \), and \( \ln 5 \). Results are approximately 0.00, 0.69, 1.10, 1.39, and 1.61, respectively. Round each value to the nearest hundredth.

Step 2: Sketch the graph

Plot the calculated values on a set of axes. Remember that \( x \)-axis units (e.g., 1 unit per major division) may differ from \( y \)-axis units (e.g., 0.5 units per major division). Connect the points to show the curve of the natural logarithm function.

Step 3: Determine the domain and range

The domain of \( f(x) = \ln x \) is all positive real numbers \((0, \infty)\) because \( x \) must be greater than zero. The range is all real numbers \((-\infty, \infty)\).

Step 4: Identify intercepts

The \( x \)-intercept occurs where \( f(x) = 0 \). Since \( \ln 1 = 0 \), the \( x \)-intercept is at \( x=1 \). The logarithm function doesn't cross the \( y \)-axis, so there is no \( y \)-intercept.

Step 5: Determine the asymptote

The graph of \( f(x) = \ln x \) has a vertical asymptote at \( x=0 \) because the logarithm approaches negative infinity as \( x \) approaches zero from the right.

Step 6: Assess if the function increases or decreases

The function \( f(x) = \ln x \) is increasing because, for any two points \( a \) and \( b \) where \( a < b \), \( \ln a < \ln b \). As \( x \) increases, \( f(x) \) increases.

Step 7: Check if the function is one-to-one

A function is one-to-one if each \( y \)-value corresponds to exactly one \( x \)-value. Since \( \ln x \) is strictly increasing and each \( y \) is paired with a unique \( x \), it is indeed a one-to-one function.

Key Concepts

Domain and Range

The domain and range are essential concepts to grasp when studying logarithmic functions. For the natural logarithm function, \( f(x) = \ln x \), understanding the domain requires recognizing the set of possible \( x \)-values that can be inputted into the function. Since the logarithm of a non-positive number is undefined, the domain of \( f(x) = \ln x \) consists solely of all positive real numbers. In interval notation, this is expressed as \((0, \infty)\).

On the other hand, when considering the range, we are looking at all the possible \( y \)-values the function can output. The natural logarithmic function can map to any real number. This means the range of \( f(x) = \ln x \) is all real numbers, denoted as \((-\infty, \infty)\). Understanding both of these aspects is crucial, as they define the behavior and possible outputs of the logarithm function.

Asymptote

An asymptote is a line that the graph of a function approaches but never actually touches. For the natural logarithm function \( f(x) = \ln x \), there is a specific type of asymptote known as a vertical asymptote. This occurs at \( x=0 \).

A vertical asymptote exists at \( x=0 \) because as \( x \) approaches zero from the right, the function \( \ln x \) tends toward negative infinity. It is important to remember that as \( x \) gets closer and closer to zero, the function values decrease without bound, but \( x \) can never actually reach 0. This behavior is key to understanding how the graph is shaped and why the function cannot take non-positive values in its domain. The existence of this asymptote helps sketch and understand the curve of \( \ln x \); it serves as a boundary to the behavior of the function.

One-to-One Function

A one-to-one function is defined as a function where each output value corresponds to exactly one input value. This means there are no repeating \( y \)-values for different \( x \)-values. The natural logarithmic function \( f(x) = \ln x \) is an excellent example of a one-to-one function.

We can determine if a function is one-to-one using the horizontal line test. For the graph of \( \ln x \), draw horizontal lines across the graph. If any horizontal line intersects the graph in more than one place, the function is not one-to-one. With \( \ln x \), you'll find that no horizontal line does so, confirming its one-to-one nature.

The increasing property of the function helps assure this one-to-one characteristic. Since \( \ln x \) is strictly increasing, as \( x \) increases, \( f(x) \) also increases. This correlation creates a unique output for every input, a hallmark of a one-to-one function. These functions have inverse functions, where for \( \ln x \), its inverse is the exponential function \( e^x \). Understanding the one-to-one property is crucial when dealing with solving equations and working with inverse operations.