Problem 1 - Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

Answer: The four steps to follow when determining the domain of a function and graphing it are: 1. Understand the concept of a function's domain. 2. Identify possible restrictions to the domain. 3. Determine the domain of the function by writing down the intervals where the function is defined and combining them. 4. Graph the function considering its domain, paying close attention to any discontinuities or undefined behavior.

Step 1: Understand the concept of a function's domain

The domain of a function \(f\) is the set of all real numbers \(x\) for which the function \(f(x)\) is defined. In other words, the domain is the set of all possible input values for the function.

Step 2: Identify possible restrictions to the domain

There can be several reasons why a function could be undefined at certain points. Some common restrictions to the domain of a function include: 1. Division by zero: Functions with a denominator must not have a zero in the denominator, as division by zero is undefined. 2. Square roots (or other even-index roots) of negative numbers: For real-valued functions, even-index roots of negative numbers are not defined, as they result in complex numbers. 3. Natural logarithmic functions: The natural logarithm of a non-positive number is undefined. 4. Other specific restrictions: Some functions may have other specific restrictions depending on their definition.

Step 3: Determine the domain of the function

To determine the domain of a function \(f\), follow these steps: 1. Identify any possible restrictions from Step 2. 2. Write down the intervals of \(x\) where the function is defined. 3. Combine these intervals to find the overall domain of the function.

Step 4: Graph the function considering its domain

Once the domain has been determined, graph the function using the information gained from the domain. The domain will help you identify any discontinuities, points of undefined behavior, or areas where the function does not exist. Pay close attention to the behavior of the function at the edges of the domain and in between any intervals of the domain.