Problem 6 - 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 \(f(x)=\ln x .\) 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? $$ \begin{array}{|c|c|} \hline \boldsymbol{x} & \boldsymbol{f}(\boldsymbol{x}) \\ \hline 0.5 & \\ 1 & \\ 2 & \\ 4 & \\ 6 & \\ 8 & \\ 10 & \\ \hline \end{array} $$ (Graph can't copy)

Step-by-Step Solution

Answer

1. \(\ln x\) values calculated. 2. Domain: \((0, \infty)\); Range: \((- fty, \infty)\). 3. \(x\)-intercept: (1,0); No \(y\)-intercept. 4. Asymptote at \(x=0\). 5. Function is increasing and one-to-one.

Step 1: Calculate Values of f(x)

Use a calculator to find the natural logarithm for each value of \(x\). Round the results to the nearest hundredth.\[\begin{align*}\ln(0.5) & \approx -0.69, \\ln(1) & = 0.00, \\ln(2) & \approx 0.69, \\ln(4) & \approx 1.39, \\ln(6) & \approx 1.79, \\ln(8) & \approx 2.08, \\ln(10) & \approx 2.30.\end{align*}\]

Step 2: Graph the Function

Plot the points \((0.5, -0.69), (1, 0.00), (2, 0.69), (4, 1.39), (6, 1.79), (8, 2.08), (10, 2.30)\) on a graph. Draw a smooth curve through these points to represent the function \(f(x)=\ln x\).

Step 3: Determine the Domain and Range

The domain of \(f(x) = \ln x\) includes all positive real numbers, \((0, \infty)\). The range includes all real numbers, \((-fty, \infty)\).

Step 4: Identify the Intercepts

The \(x\)-intercept is at the point where \(f(x)=0\). From the table, \(f(1) = 0\), so the \(x\)-intercept is \((1, 0)\). The function \(f(x)=\ln x\) does not have a \(y\)-intercept because it is undefined for \(x=0\).

Step 5: Identify the Asymptote

The function \(f(x) = \ln x\) has a vertical asymptote at \(x = 0\). As \(x\) approaches 0 from the positive side, \(f(x)\) approaches negative infinity.

Step 6: Analyze the Behavior

The function \(f(x) = \ln x\) is an increasing function. As \(x\) increases, \(f(x)\) also increases.

Step 7: Determine if the Function is One-to-One

The function \(f(x) = \ln x\) is one-to-one. For each unique input \(x\), there is a unique output \(f(x)\).

Key Concepts

Domain and Range

Understanding the domain and range of functions is crucial, particularly in the case of the natural logarithm function, denoted as \(f(x) = \ln x\). For this function, the **domain** is all positive real numbers, written as \((0, \infty)\). This means that \(x\) must be greater than 0, as you cannot take the logarithm of a non-positive number. Remember: both positive fractions and integers can be inputs for \(f(x)\), but zero and negatives cannot. On the other hand, the **range** of \(f(x) = \ln x\) covers all real numbers, \((-\infty, \infty)\). This means that as \(x\) takes on values greater than zero, the outputs \(f(x)\) can be any real number. As \(x\) gets larger, \(f(x)\) increases without bound, and as \(x\) gets closer to zero, \(f(x)\) becomes a larger negative number.

Graphing Functions

Graphing the natural logarithm function involves plotting points and connecting them with a smooth curve. The function \(f(x) = \ln x\) is unique due to its characteristic shape. To graph this function, consider the points obtained from calculated values such as \((0.5, -0.69)\), \((1, 0.00)\), and \((2, 0.69)\). These points reflect the function's behavior as it increases slowly at first and then more rapidly as \(x\) increases.When creating the graph, notice the key characteristics:

• The curve passes through the point \((1, 0)\), indicating that when \(x = 1\), the logarithm of 1 is zero.
• There is no \(y\)-intercept because \(\ln x\) is undefined for \(x = 0\).

Drawing the curve requires smoothly connecting these points, remembering that the function approaches negative infinity as \(x\) approaches zero from the positive side.

Asymptotes

An asymptote is a line that a graph approaches but never touches. For the natural logarithm function \(f(x) = \ln x\), a vertical asymptote exists at \(x = 0\). As \(x\) approaches zero from the positive side, \(f(x)\) decreases without bound, heading towards negative infinity. Thus,

• The graph gets infinitely close to the \(y\)-axis without ever touching or crossing it.
• This vertical asymptote is a critical feature of the \(\ln x\) graph, signifying the function's undefined nature at zero and how it behaves for very small \(x\) values.

Understanding asymptotes helps in recognizing how the graph behaves as it extends to extreme values.

One-to-One Functions

A function is considered one-to-one if each \(x\)-value corresponds to a unique \(y\)-value, and vice versa. The natural logarithm function \(f(x) = \ln x\) is a classic example. Being one-to-one means that:

• If \(f(a) = f(b)\), then there are equal inputs: \(a = b\).
• No two different \(x\)-values will produce the same \(f(x)\).

This characteristic is important for determining an inverse function. Since \(f(x) = \ln x\) is one-to-one, it has a unique inverse, which is the exponential function \(e^x\). One-to-one functions like the natural logarithm are foundational in understanding how inverses and various transformations work, especially within calculus and advanced algebra.