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Domain and Range of a Function Domain and Range of a Function Andrew Gloag Eve Rawley Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) AUTHORS Andrew Gloag To access a customizable version of this book, as well as other Eve Rawley interactive content, visit www.ck12.org SOURCE Anne Gloag CK-12 Foundation is a non-profit organization with a mission to reduce the cost of textbook materials for the K-12 market both in the U.S. and worldwide. Using an open-source, collaborative, and web-based compilation model, CK-12 pioneers and promotes the creation and distribution of high-quality, adaptive online textbooks that can be mixed, modified and printed (i.e., the FlexBook® textbooks). Copyright © 2015 CK-12 Foundation, www.ck12.org The names “CK-12” and “CK12” and associated logos and the terms “FlexBook®” and “FlexBook Platform®” (collectively “CK-12 Marks”) are trademarks and service marks of CK-12 Foundation and are protected by federal, state, and international laws. Any form of reproduction of this book in any format or medium, in whole or in sections must include the referral attribution link http://www.ck12.org/saythanks (placed in a visible location) in addition to the following terms. Except as otherwise noted, all CK-12 Content (including CK-12 Curriculum Material) is made available to Users in accordance with the Creative Commons Attribution-Non-Commercial 3.0 Unported (CC BY-NC 3.0) License (http://creativecommons.org/ licenses/by-nc/3.0/), as amended and updated by Creative Com- mons from time to time (the “CC License”), which is incorporated herein by this reference. Complete terms can be found at http://www.ck12.org/about/ terms-of-use. Printed: December 13, 2015 www.ck12.org Chapter 1. Domain and Range of a Function CHAPTER 1 Domain and Range of a Function Here you’ll learn how to find the domain and range of a function and you’ll make a table of values for a given function. What if you were given a rule that relates two variables, like f (x) = 5x2 + 1? How could you find the domain and range of the function defined by that rule? After completing this Concept, you’ll be able to identify the domain and range of functions like this one. Watch This MEDIA Click image to the left or use the URL below. URL: http://www.ck12.org/flx/render/embeddedobject/133266 CK-12 Foundation: 0110S Introduction to Functions For another look at the domain of a function, see the following video, where the narrator solves a sample problem from the California Standards Test about finding the domain of an unusual function. MEDIA Click image to the left or use the URL below. URL: http://www.ck12.org/flx/render/embeddedobject/5 Khan Academy: CA AlgebraI: Functions Guidance A function is a rule for relating two or more variables. For example, the price you pay for phone service may depend on the number of minutes you talk on the phone. We would say that the cost of phone service is a function of the number of minutes you talk. Consider the following situation. Josh goes to an amusement park where he pays $2 per ride. There is a relationship between the number of rides Josh goes on and the total amount he spends that day: To figure out the amount he spends, we multiply the number of rides by two. This rule is an example of a function. Functions usually—but not always—are rules based on mathematical operations. You can think of a function as a box or a machine that contains a mathematical operation. 1 www.ck12.org Whatever number we feed into the function box is changed by the given operation, and a new number comes out the other side of the box. When we input different values for the number of rides Josh goes on, we get different values for the amount of money he spends. The input is called the independent variable because its value can be any number. The output is called the dependent variable because its value depends on the input value. Functions usually contain more than one mathematical operation. Here is a situation that is slightly more complicated than the example above. Jason goes to an amusement park where he pays $8 admission and $2 per ride. The following function represents the total amount Jason pays. The rule for this function is "multiply the number of rides by 2 and add 8." When we input different values for the number of rides, we arrive at different outputs (costs). These flow diagrams are useful in visualizing what a function is. However, they are cumbersome to use in practice. In algebra, we use the following short-hand notation instead: # f (x) = y outputfunctionbox |{z} First, we define the variables: x = the number of rides Jason goes on y = the total amount of money Jason spends at the amusement park. So, x represents the input and y represents the output. The notation f () represents the function or the mathematical operations we use on the input to get the output. In the last example, the cost is 2 times the number of rides plus 8. This can be written as a function: f (x) = 2x + 8 2 www.ck12.org Chapter 1. Domain and Range of a Function In algebra, the notations y and f (x) are typically used interchangeably. Technically, though, f (x) represents the function itself and y represents the output of the function. Identify the Domain and Range of a Function In the last example, we saw that we can input the number of rides into the function to give us the total cost for going to the amusement park. The set of all values that we can use for the input is called the domain of the function, and the set of all values that the output could turn out to be is called the range of the function. In many situations the domain and range of a function are both simply the set of all real numbers, but this isn’t always the case. Let’s look at our amusement park example. Example A Find the domain and range of the function that describes the situation: Jason goes to an amusement park where he pays $8 admission and $2 per ride. Solution Here is the function that describes this situation: f (x) = 2x + 8 = y In this function, x is the number of rides and y is the total cost. To find the domain of the function, we need to determine which numbers make sense to use as the input (x). • The values have to be zero or positive, because Jason can’t go on a negative number of rides. • The values have to be integers because, for example, Jason could not go on 2.25 rides. • Realistically, there must be a maximum number of rides that Jason can go on because the park closes, he runs out of money, etc. However, since we aren’t given any information about what that maximum might be, we must consider that all non-negative integers are possible values regardless of how big they are. Answer For this function, the domain is the set of all non-negative integers. To find the range of the function we must determine what the values of y will be when we apply the function to the input values. The domain is the set of all non-negative integers: {0, 1, 2, 3, 4, 5, 6, ...}. Next we plug these values into the function for x. If we plug in 0, we get 8; if we plug in 1, we get 10; if we plug in 2, we get 12, and so on, counting by 2s each time. Possible values of y are therefore 8, 10, 12, 14, 16, 18, 20... or in other words all even integers greater than or equal to 8. Answer The range of this function is the set of all even integers greater than or equal to 8. Example B Find the domain and range of the following functions. a) A ball is dropped from a height and it bounces up to 75% of its original height. b) y = x2 Solution a) Let’s define the variables: x = original height y = bounce height 3 www.ck12.org A function that describes the situation is y = f (x) = 0:75x. x can represent any real value greater than zero, since you can drop a ball from any height greater than zero. A little thought tells us that y can also represent any real value greater than zero. Answer The domain is the set of all real numbers greater than zero. The range is also the set of all real numbers greater than zero. b) Since there is no word problem attached to this equation, we can assume that we can use any real number as a value of x. When we square a real number, we always get a non-negative answer, so y can be any non-negative real number. Answer The domain of this function is all real numbers. The range of this function is all non-negative real numbers. In the functions we’ve looked at so far, x is called the independent variable because it can be any of the values from the domain, and y is called the dependent variable because its value depends on x. However, any letters or symbols can be used to represent the dependent and independent variables. Here are three different examples: y = f (x) = 3x R = f (w) = 3w v = f (t) = 3t These expressions all represent the same function: a function where the dependent variable is three times the independent variable.
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