Problem 20 - A Graphical Approach to Precalculus with Limits

Question

Explain how the graph of \(f\) can be obtained from the graph of \(y=\frac{1}{x}\) or \(y=\frac{1}{x^{2}} .\) Draw a sketch of the graph of \(f\) by hand. Then generate an accurate depiction of the graph with a graphing calculator. Finally, give the domain and range. $$f(x)=\frac{1}{x-3}$$

Step-by-Step Solution

Answer

The graph is the shifted hyperbola \(\frac{1}{x}\), moved 3 units right; domain: \(x \neq 3\), range: \(y \neq 0\).

Step 1: Identifying the Base Graph

To understand how the graph of \(f(x) = \frac{1}{x-3}\) can be derived, we first identify the base function. In this case, the base function is \(y = \frac{1}{x}\), known as the reciprocal function, which has a hyperbola shape centered at the origin.

Step 2: Understanding the Transformation

The function \(f(x) = \frac{1}{x-3}\) represents a horizontal shift of the graph \(y = \frac{1}{x}\). Specifically, the term \(x-3\) means that the hyperbola is shifted 3 units to the right. This changes the vertical asymptote from \(x = 0\) to \(x = 3\).

Step 3: Sketching the Graph by Hand

Start by sketching the basic \(y = \frac{1}{x}\) hyperbola with asymptotes at \(x = 0\) and \(y = 0\). Then shift every point on the graph 3 units to the right. The asymptotes move to \(x = 3\) and \(y = 0\). Draw the two branches of the hyperbola approaching these asymptotes.

Step 4: Using a Graphing Calculator

To get an accurate graph, enter \(f(x) = \frac{1}{x-3}\) into the graphing calculator. Observe that the graph matches your hand-drawn sketch, with branches in the first and third quadrants, asymptotic to \(x = 3\) and \(y = 0\).

Step 5: Determining the Domain

The domain of \(f(x) = \frac{1}{x-3}\) is all real numbers except where the denominator is zero. Since \(x-3 = 0\) when \(x = 3\), the domain is all real numbers \(x\) such that \(x eq 3\).

Step 6: Determining the Range

The range of \(f(x) = \frac{1}{x-3}\) is all real numbers except where the graph approaches infinity, which is never reached. Thus, there is no horizontal asymptote. The range is all real numbers \(y\) except \(y eq 0\).

Key Concepts

Reciprocal Function

A reciprocal function is generally of the form \( y = \frac{1}{x} \) or \( y = \frac{1}{x^n} \). These functions are characterized by their hyperbolic graphs. They always have two asymptotes: a vertical asymptote (usually at the line \( x = 0 \)) and a horizontal asymptote (usually at the line \( y = 0 \)). These asymptotes are lines that the graph approaches but never touches.

In the case of the reciprocal function \( y = \frac{1}{x} \), the graph is a hyperbola with branches in the first and third quadrants. This results from the positive and negative values of \( x \), respectively, yielding positive and negative values for \( y \). The properties of the reciprocal function provide a basis for understanding transformations that can be applied to it, such as shifts and stretches.

• The base graph has symmetry about the origin.
• As \( x \to 0^+ \) or \( x \to 0^- \), \( y \to \infty \) or \( y \to -\infty \).
• As \( x \to \infty \) or \( x \to -\infty \), \( y \to 0 \).

Horizontal Shifts

In functions, horizontal shifts are transformations that involve moving the graph left or right along the x-axis. For a function like \( f(x) = \frac{1}{x-3} \), the horizontal shift can be identified by looking at the expression inside the reciprocal term. Here, \( x-3 \) suggests a shift to the right by 3 units.

The effect of this horizontal shift changes more than just the graph's position. It also alters the location of the vertical asymptote. Originally at \( x = 0 \) in the base reciprocal function, the new vertical asymptote for \( f(x) = \frac{1}{x-3} \) shifts to \( x = 3 \). The horizontal asymptote remains unchanged at \( y = 0 \).

• Each point on the graph moves rightward by 3 units.
• The transformation is visualized by replacing \( x \) with \( x - 3 \) in the function.
• The function's general shape remains as a hyperbola, simply moved to a new center.

Domain and Range

Understanding the domain and range of a function is key to fully grasping its behavior on a graph. For \( f(x) = \frac{1}{x-3} \), the domain and range provide insight into the values that the function can accept or produce.

Starting with the domain, which is the set of all possible x-values that can be input into the function, we look to avoid division by zero. This is because division by zero results in an undefined value. Therefore, for \( x = 3 \), the denominator will be zero, which means \( x eq 3 \). Thus, the domain is all real numbers except 3.

• Those familiar with set notation might express this as \( (-\infty, 3) \cup (3, \infty) \).

The range of a function refers to the possible y-values that can be output by the function. For \( f(x) = \frac{1}{x-3} \), the function never reaches \( y = 0 \) because the graph approaches the x-axis but never touches it. Therefore, \( y eq 0 \).

• The full range can also be written as \( (-\infty, 0) \cup (0, \infty) \).