3. Functions COMMON CORE Learning Standard HSF-IF.A. Essential Question What is a function? A relation pairs inputs with outputs. When a relation is given as ordered pairs, the -coordinates are inputs and the -coordinates are outputs. A relation that pairs each input with eactl one output is a function. ANALYZING RELATIONSHIPS To be proficient in math, ou need to analze relationships mathematicall to draw conclusions. Describing a Function Work with a partner. Functions can be described in man was. b an equation b an input-output table using words b a graph as a set of ordered pairs a. Eplain wh the graph shown represents a function. b. Describe the function in two other was. Identifing Functions Work with a partner. Determine whether each relation represents a function. Eplain our reasoning. 8 8 a. Input, 3 Output, 8 8 8 8 8 b. Input, 8 8 8 8 8 Output, 3 c. Input, Output, d. 3 8 9 8 8 e. (, 5), (, 8), (, ), (, ), (, 7) f. (, ), (, ), (, ), (, ), (, ), (, ) g. Each radio frequenc in a listening area has eactl one radio station. h. The same television station can be found on more than one channel. i. = j. = + 3 Communicate Your Answer 3. What is a function? Give eamples of relations, other than those in Eplorations and, that (a) are functions and (b) are not functions. Section 3. Functions 3
3. Lesson Core Vocabular relation, p. function, p. domain, p. range, p. independent variable, p. 7 dependent variable, p. 7 Previous ordered pair mapping diagram What You Will Learn Determine whether relations are functions. Find the domain and range of a function. Identif the independent and dependent variables of functions. Determining Whether Relations Are Functions A relation pairs inputs with outputs. When a relation is given as ordered pairs, the -coordinates are inputs and the -coordinates are outputs. A relation that pairs each input with eactl one output is a function. Determining Whether Relations Are Functions Determine whether each relation is a function. Eplain. a. (, ), (, ), (, ), (, ), (, ) b. (, ), (8, 7), (, ), (, 3), (5, ) c. Input, REMEMBER A relation can be represented b a mapping diagram. d. Output, 3 5 7 8 Input, 3 Output, 5 SOLUTION a. Ever input has eactl one output. So, the relation is a function. b. The input has two outputs, and 3. So, the relation is not a function. c. The input has two outputs, 5 and. So, the relation is not a function. d. Ever input has eactl one output. So, the relation is a function. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Determine whether the relation is a function. Eplain.. ( 5, ), (, ), (5, ), (5, ). (, 8), (, ), (, ), (5, ) 3. Input, Output,. Input, Output,. 5. 7.8 Chapter 3 Graphing Linear Functions
Core Concept Vertical Line Test Words A graph represents a function when no vertical line passes through more than one point on the graph. Eamples Function Not a function Using the Vertical Line Test Determine whether each graph represents a function. Eplain. a. b. SOLUTION a. You can draw a vertical line through (, ) and (, 5). So, the graph does not represent a function. b. No vertical line can be drawn through more than one point on the graph. So, the graph represents a function. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Determine whether the graph represents a function. Eplain. 5.. 7. 8. Section 3. Functions 5
Finding the Domain and Range of a Function Core Concept The Domain and Range of a Function The domain of a function is the set of all possible input values. The range of a function is the set of all possible output values. input output Finding the Domain and Range from a Graph Find the domain and range of the function represented b the graph. a. b. 3 3 STUDY TIP A relation also has a domain and a range. SOLUTION a. Write the ordered pairs. Identif the inputs and outputs. inputs ( 3, ), (, ), (, ), (3, ) outputs The domain is 3,,, and 3. The range is,,, and. b. Identif the - and -values represented b the graph. 3 3 domain range The domain is 3. The range is. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Find the domain and range of the function represented b the graph. 9.. Chapter 3 Graphing Linear Functions
Identifing Independent and Dependent Variables The variable that represents the input values of a function is the independent variable because it can be an value in the domain. The variable that represents the output values of a function is the dependent variable because it depends on the value of the independent variable. When an equation represents a function, the dependent variable is defined in terms of the independent variable. The statement is a function of means that varies depending on the value of. = + dependent variable, independent variable, Identifing Independent and Dependent Variables The function = 3 + represents the amount (in fluid ounces) of juice remaining in a bottle after ou take gulps. a. Identif the independent and dependent variables. b. The domain is,,, 3, and. What is the range? SOLUTION a. The amount of juice remaining depends on the number of gulps. So, is the dependent variable, and is the independent variable. b. Make an input-output table to find the range. Input, 3 + Output, 3() + 3() + 9 3() + 3 3(3) + 3 3() + The range is, 9,, 3, and. Monitoring Progress Help in English and Spanish at BigIdeasMath.com. The function a = b + represents the number a of avocados ou have left after making b batches of guacamole. a. Identif the independent and dependent variables. b. The domain is,,, and 3. What is the range?. The function t = 9m + 5 represents the temperature t (in degrees Fahrenheit) of an oven after preheating for m minutes. a. Identif the independent and dependent variables. b. A recipe calls for an oven temperature of 35 F. Describe the domain and range of the function. Section 3. Functions 7
3. Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check. WRITING How are independent variables and dependent variables different?. DIFFERENT WORDS, SAME QUESTION Which is different? Find both answers. Find the range of the function represented b the table. Find the inputs of the function represented b the table. 7 5 Find the -values of the function represented b (, 7), (, 5), and (, ). Find the domain of the function represented b (, 7), (, 5), and (, ). Monitoring Progress and Modeling with Mathematics In Eercises 3 8, determine whether the relation is a function. Eplain. (See Eample.) 3. (, ), (, ), (3, ), (, 3), (5, )... (7, ), (5, ), (3, 8), (, 5), (3, ) 5. Input, Output,. Input, Output, 3 5 7 3 3 3 8 In Eercises 3, find the domain and range of the function represented b the graph. (See Eample 3.) 3.. 7. Input, Output, 8. Input, 3 3 9 Output, 5 7 3 5.. In Eercises 9, determine whether the graph represents a function. Eplain. (See Eample.) 9.. 7. MODELING WITH MATHEMATICS The function = 5 + 5 represents our monthl rent (in dollars) when ou pa das late. (See Eample.) a. Identif the independent and dependent variables. b. The domain is,,, 3,, and 5. What is the range? 8 Chapter 3 Graphing Linear Functions
8. MODELING WITH MATHEMATICS The function = 3.5 +.8 represents the cost (in dollars) of a tai ride of miles. a. Identif the independent and dependent variables. b. You have enough mone to travel at most miles in the tai. Find the domain and range of the function. ERROR ANALYSIS In Eercises 9 and, describe and correct the error in the statement about the relation shown in the table. Input, 3 5 Output, 7 8 9. MULTIPLE REPRESENTATIONS The function.5 +.5 = represents the number of hardcover books and softcover books ou can bu at a used book sale. a. Solve the equation for. b. Make an input-output table to find ordered pairs for the function. c. Plot the ordered pairs in a coordinate plane. 5. ATTENDING TO PRECISION The graph represents a function. Find the input value corresponding to an output of.. OPEN-ENDED Fill in the table so that when t is the independent variable, the relation is a function, and when t is the dependent variable, the relation is not a function. 9. The relation is not a function. One output is paired with two inputs. t v. The relation is a function. The range is,, 3,, and 5. 7. ANALYZING RELATIONSHIPS You select items in a vending machine b pressing one letter and then one number. ANALYZING RELATIONSHIPS In Eercises and, identif the independent and dependent variables.. The number of quarters ou put into a parking meter affects the amount of time ou have on the meter.. The batter power remaining on our MP3 plaer is based on the amount of time ou listen to it. 3. MULTIPLE REPRESENTATIONS The balance (in dollars) of our savings account is a function of the month. Month, 3 Balance (dollars), 5 5 75 a. Describe this situation in words. b. Write the function as a set of ordered pairs. c. Plot the ordered pairs in a coordinate plane. a. Eplain wh the relation that pairs letter-number combinations with food or drink items is a function. b. Identif the independent and dependent variables. c. Find the domain and range of the function. Section 3. Functions 9
8. HOW DO YOU SEE IT? The graph represents the height h of a projectile after t seconds. Height (feet) Height of a Projectile 5 h 5 5.5.5.5 t Time (seconds) a. Eplain wh h is a function of t. b. Approimate the height of the projectile after.5 second and after.5 seconds. c. Approimate the domain of the function. d. Is t a function of h? Eplain. 3. A function pairs each chaperone on a school trip with students. REASONING In Eercises 35 38, tell whether the statement is true or false. If it is false, eplain wh. 35. Ever function is a relation. 3. Ever relation is a function. 37. When ou switch the inputs and outputs of an function, the resulting relation is a function. 38. When the domain of a function has an infinite number of values, the range alwas has an infinite number of values. 39. MATHEMATICAL CONNECTIONS Consider the triangle shown. 9. MAKING AN ARGUMENT Your friend sas that a line alwas represents a function. Is our friend correct? Eplain. h 3 3. THOUGHT PROVOKING Write a function in which the inputs and/or the outputs are not numbers. Identif the independent and dependent variables. Then find the domain and range of the function. ATTENDING TO PRECISION In Eercises 3 3, determine whether the statement uses the word function in a wa that is mathematicall correct. Eplain our reasoning. 3. The selling price of an item is a function of the cost of making the item. 3. The sales ta on a purchased item in a given state is a function of the selling price. 33. A function pairs each student in our school with a homeroom teacher. Maintaining Mathematical Proficienc Write the sentence as an inequalit. (Section.) a. Write a function that represents the perimeter of the triangle. b. Identif the independent and dependent variables. c. Describe the domain and range of the function. (Hint: The sum of the lengths of an two sides of a triangle is greater than the length of the remaining side.) REASONING In Eercises 3, find the domain and range of the function.. =. =. = 3. =. A number is less than. 5. Three is no less than a number.. Seven is at most the quotient of a number d and 5. 7. The sum of a number w and is more than. Reviewing what ou learned in previous grades and lessons Evaluate the epression. (Skills Review Handbook) 8. 9. ( 3) 5. 5 5. 5 Chapter 3 Graphing Linear Functions