194 Chapter 3 Graphs, Relations, and Functions Step 2: If the function has a fraction with a variable in the denominator, set the denominator equal to zero to find the value(s) that make(s) the function undefined. The set of real numbers excluding these values is the domain of the function. Step 3: If the function involves the square root of an expression, set the expression inside the square root greater than or equal to zero to find its domain. The solution of the inequality is the domain of the function. Objective 3 Examples Find the domain of each function from its equation. 3a. f (x) = 3x - 5 3b. f (x) = 2 x - 3 3c. f (x) = 1x - 4 Solutions 3a. The expression 3x 2 5 is defined for all real numbers. So, the domain is (-∞, ∞). 3b. The expression 2 x 2 3 is undefined when x 2 3 5 0 or when x 5 3. The domain is all real numbers except x 5 3. We write this as (-∞, 3) ∪ (3, ∞). 3c. The expression !x 2 4 is defined only when x 2 4 $ 0 or x $ 4. We write the domain as 4, ∞). Student Check 3 Find the domain of each function from its equation. a. f (x) = 2x + 1 b. f (x) = 4 x + 7 c. f (x) = 1x - 2 Applications We have already established that functions occur in many real-world situations. When a student uses a student ID number to register for classes, the number is paired with the student name so that the student gets registered for the appropriate classes. The domain of this function is the set of student ID numbers and the range is the set of students assigned to an ID. Another example of a function involves e-mail. When a person e-mails someone, an e-mail address is paired with a particular recipient. The domain in this function is the e-mail address and the range is the recipient. The list could go on with these types of examples. In each of these situations, the domain and range of these functions has to be defined. Someone has to set up restrictions on what is going to be a reasonable value for the domain (input) and the range (output) of the function. In Example 4, we are in charge of determining an appropriate domain of a particular function. These examples come from problems that will be discussed later in the book. Objective 4 Examples Determine an appropriate domain for each situation. 4a. The revenue R for selling d donuts a month for a bakery is given by R(d) = 0.65d - 4500 4b. The area of the rectangle can be represented by the function A(x) = x(10 - 2x). x 10 – 2x Objective 4 ▶ Apply the concept of domain to real-world situations.
hendricks_intermediate_algebra_1e_ch1_3
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