Problem 2 - Calculus Early Transcendentals

Question

Why is it important to determine the domain of \(f\) before graphing \(f ?\)

Step-by-Step Solution

Answer

Answer: Determining the domain of a function before graphing is important for several reasons, including preventing mathematical errors, guiding the range of x-values to be considered, and identifying the behavior of the function at boundaries or discontinuities. This ensures the accurate representation of the function's graph and helps avoid inaccuracies or misinterpretations.

Step 1: Define Domain

The domain of a function \(f(x)\) is the set of all possible input values (x-values) for which the function is defined. These input values should result in a unique output value (y-value) and not cause any mathematical errors or contradictions.

Step 2: Discuss the Importance of Domain

Determining the domain of a function before graphing is critical because of the following reasons: 1. It helps to prevent mathematical errors, such as division by zero or taking the square root of a negative number. 2. It guides the range of x-values to be considered when graphing the function. 3. Knowing the domain also aids in identifying the behavior of the function at the boundaries or any discontinuities that may occur.

Step 3: Domain and Graphing Example

To illustrate the importance of determining the domain before graphing, let's consider the function \(f(x)=\frac{1}{x-3}\). First, find the domain of \(f(x)\). To do so, determine for which values of \(x\) the function is defined: \(f(x) = \frac{1}{x-3}\) is undefined only when the denominator is equal to zero: \(x-3=0\) Solving for \(x\), we find that \(x=3\). Thus, the domain of \(f(x)\) is all real numbers except \(x=3\). In interval notation, this is \((-\infty, 3) \cup (3, \infty)\). Now, when graphing this function, knowing the domain helps us avoid any errors or inaccuracies as we know that the function cannot be defined for \(x=3\). This information also informs us that there is a vertical asymptote at \(x=3\), which is an essential feature of the graph.