Week 2 Topic 4 Domain and Range 1 Week 2 Topic 4 Domain and Range Introduction A function is a rule that takes an input and produces one output. If we want functions to represent real-world situations, we have to know what the possibilities are for input (domain) and output (range). In this section, we look at finding domain and range from equations and tables. We ll explore graphs in a later section. This can be a difficult concept because there is no formulaic way to find domain and range each function is different. But domain is much easier if you remember a couple facts: 1. Denominators can t be equal to zero 2. You can t take the square root of a negative number (or fourth root, sixth root, etc.) Key Points 1. What is the domain of a function? How do you find domain from a table or equation? 2. What is the range of a function? How do you find range from a table? 3. When working with an equation, it s usually easier to find what numbers are NOT in the domain. Vocabulary & Definitions The domain of a function is the set of inputs The range of a function is the set of outputs This definition says domain and range are always sets, but intervals are okay too. Know how to use both. Reading Set notation review next page Understanding Domain after that Textbook pages 48 51
Week 2 Topic 4 Domain and Range 2 Set Notation Review A set is a collection of objects, like numbers, enclosed in curly braces { }. A set is like a suitcase for numbers; it lets you carry them around from place to place. For example, {1, 2, 3} is the set of the numbers 1, 2, and 3. These numbers are elements of the set. Uses of Sets We can use sets to denote solutions of equations. For example, the equation x 2 4 has solutions x 2 and x -2. We say the solution set is {-2, 2}. That s a little less to write. We can also use sets to represent large quantities of numbers. For example, the set of all positive even numbers is {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, }. There are infinitely many even numbers, so we can t list them all, but we can use the ellipsis ( ) to indicate the pattern continues. What if we want a set that contains every number bigger than 1? That would include 2, 3, 4, but also 1.5, 1.55, 1.555,. There s no way to use a single to represent all of the numbers! However, we can use set-builder notation to write a formula for the numbers in a set without listing every number. The set of numbers bigger than 1 can be written in set notation as {x x > 1} This reads The set of x such that x is greater than 1 It means All numbers are included, if the number is bigger than 1 The vertical bar means such that in math lingo. You can find it above the Enter key on most keyboards. Every number greater than 1 is in this set. We haven t listed any numbers here, just written a formula for what numbers are in the set. This is much more convenient than trying to list them! Set Problems 1. Write sets for a. Numbers greater than or equal to zero b. Numbers not equal to -5
Week 2 Topic 4 Domain and Range 3 Understanding Domain A function defined by a formula is a rule for how to get from the input to the output. This rule sometimes comes with conditions. For example, if I drop a ball off a building 50 meters high, the height of the ball at time t (in seconds) is: h(t) 50 4.9t 2 This function only makes sense if t 0, and then only until the ball hits the ground. This function has conditions, and we might write it as h(t) 50 4.9t 2 0 t 3.194 Other times, we might see the formula and have to figure out what conditions make sense for ourselves. The conditions on what we can put into the function give us the domain of the function. For example, we could define the function f as f ( x) x Now we have to notice that this formula only works if x 0. Any positive number is fine, but we can t input a negative. We could write an interval for the domain as [0, ) or a set {x x 0}. We could define the function g as g( x) x for x 1. This function s domain is [1, ) or {x x 1} The difference between f and g is that g comes with a condition already in place, and for f we have to use our reasoning ability to figure out the domain It s important to note that f and g have the same formula, but they are different functions because they have different conditions and therefore different domains. In this class, you might see functions either way. In most math classes, you have to look for a few things: 1. Is the domain given to you in the problem? 2. If it s a word problem, what values of the variable make sense in the problem? 3. Are there implicit conditions like denominators or square roots that impose conditions without being stated directly?
Week 2 Topic 4 Domain and Range 4 Most cases in this class will be number 3 from above. In that case, finding the domain of a function from an equation is a matter of finding which numbers don t work. Formulas with Square Roots To find the domain of an equation with a square root, you need to find where the radicand (what s under the radical sign) is greater than or equal to zero. Example: Find the domain of f( x) x + 1. Solution: You can t square root a negative: x + 1 0. Solve the inequality for x: x -1 In set notation: {x x -1} In interval notation: [-1, ) Formulas with Fractions To find the domain of an equation with a fraction, you need to find where the denominator is not equal to zero. 1 Example: Find the domain of f( x). x + 7 Solution: The denominator can t be equal to zero: x + 7 0 Solve for x: x -7 So x can be any number other than 1. In set notation: {x x -7} In interval notation: (-, -7) (-7, ) Note: The is a union symbol: it means to include everything in both intervals. Conclusions The range of a function is the set of all possible outputs. Finding the range is harder without knowing the function s behavior, so we focus on finding the domain for the time being. Every time we learn a new type of function (exponentials, logarithms, etc.), we learn the function s domain and range as well. It s worth your time to make sure you understand domain and range. On exams, the average score for these problems is often under 40% even though there are only a couple things to remember.
Week 2 Topic 4 Domain and Range 5 Domain and Range Concept Exercises 1. Find the domain and range of the functions defined in these tables. Write answers using set notation. x 1 2 3 4 x -2-1 0 1 F(x) 2 2 3 3 g(x) 4 5 6 7 Domain: Domain: Range: Range: 2. Find the domain of each of these functions. Write answers in set notation and interval notation. a. h( x) 2 x b. T( r) 2 2r + 1 3. Find the domain of each of these functions. Write answers in set notation and interval notation. a. 1 H( s) 2s 1 b. 10 x F( x) 7 2x 4. For equations with both radicals and fractions, find where the radicand is 0 and where the denominator 0. Find the domain of the function f( x) 2 x 3x 1. Write your answer in both set notation and interval notation.
Week 2 Topic 4 Domain and Range 6 Answers to 2: a. {x x 2} b. [-1/2, ) Answers to 3: a. {s s ½} b. (-, 3.5) (3.5, ) Answer to 4: {x x > 1/3}