Problem 5 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Sketch the graph of the function without the use of a computer or graphing calculator. $$ y=\ln \left(\frac{1}{x}\right) $$

Step-by-Step Solution

Answer

The graph of the function \( y = \ln \left(\frac{1}{x}\right) \) starts at the point (-1,0) and decreases without bound as x decreases. The domain of the function is all non-zero real numbers (-Infinity,0) union (0, +Infinity) and the range of the function is all real numbers.

Step 1: Identify Basic Graph of Logarithmic Function

The basic graph of the function \( y = \ln(x) \) starts at the point (1,0) and increases without bound as x increases, since any positive number can be made by raising e to some power. Logarithms are not defined for negative values of x, hence, the basic graph does not extend into the left half of the x-y plane.

Step 2: Apply Transformation

The transformation due to \(\frac{1}{x}\) causes a reflection of the graph over the y-axis. This results in the function \( y = \ln \left(\frac{1}{x}\right) \) which starts at the point (-1,0) and decreases without bound as x decreases.

Step 3: Define Domain and Range

Given the nature of logarithms, the function \( y = \ln \left(\frac{1}{x}\right) \) will be undefined for x=0. Therefore, the domain is all non-zero real numbers, or (-Infinity, 0) union (0, +Infinity). The range of the function is all real numbers.

Step 4: Sketch Final Graph

Combine all the information above to sketch the graph of \( y = \ln \left(\frac{1}{x}\right) \). The graph should start around (-1,0), show a decrease as x decreases, and show no graph for x=0. The graph will span the entire y-axis.

Key Concepts

Transformation of Functions

Understanding the transformation of functions is a foundational skill in graphing. It involves shifting, scaling, reflecting, or rotating the graph of a basic function to create a new function.

When we look at the function given by the equation \( y = \text{ln}\left(\frac{1}{x}\right) \), a key transformation is the reflection over the y-axis. This occurs due to the \( \frac{1}{x} \) inside the logarithm. To recognize this, consider the reciprocal function \( f(x) = \frac{1}{x} \), which flips the sign of x before it is processed by the log function. In terms of axes, imagine every point on the graph of \( y = \text{ln}(x) \) being mirrored to the opposite side of the y-axis.

Logarithmic Function Properties

Logarithmic functions have distinct properties that affect their graphs. One key property is their domain: logarithms are only defined for positive real numbers. Therefore, the graph of a log function, like \( y = \text{ln}(x) \), does not exist for \( x \leq 0 \).

Furthermore, as x approaches zero, the function values approach negative infinity, which is represented by a vertical asymptote on the graph. The range of logarithmic functions is all real numbers since the output can be any value between negative and positive infinity.

Remember, logarithmic functions are also inverses of exponential functions. This inverse relationship is why the graph of \( y = \text{ln}(x) \) increases without bound as x increases – paralleling the exponential function's unbounded growth.

Domain and Range of a Function

The domain of a function consists of all the input values (x values) for which the function is defined, whereas the range is the set of all possible output values (y values).

For the function \( y = \text{ln}\left(\frac{1}{x}\right) \), the domain is all real numbers except for zero, since you cannot take the logarithm of zero or a negative number. This can be expressed as \( x \in (-\infty, 0) \cup (0, +\infty) \). The range, on the other hand, is \( y \in (-\infty, +\infty) \), reflecting the fact that logarithmic functions can produce any real number as an output.

One important piece of advice for determining domain and range is to consider the limitations imposed by the operations within the function. For instance, because dividing by zero is undefined, it consequently impacts the domain of our function.

Sketching Graphs

Sketching graphs is a visual way to represent the behavior of functions. A good sketch requires understanding the function's properties, transformations, domain, and range.

For logarithmic functions like \( y = \text{ln}\left(\frac{1}{x}\right) \), start with the basic log function and apply any transformations. Next, indicate the vertical asymptote at \( x = 0 \) since the logarithm is not defined there. As \( x \) approaches 0 from the right, the function heads towards negative infinity, forming a sharp decline in the graph.

Finally, plot a few key points to ensure accuracy, such as the reflection of the point (1,0) to (-1,0). Remember, the graph should never touch the vertical asymptote and should extend infinitely along the y-axis, showing the range of all real numbers.