Function Notation. Essential Question How can you use function notation to represent a function?

Transcription

1 . Function Notation Essential Question How can ou use function notation to represent a function? The notation f(), called function notation, is another name for. This notation is read as the value of f at or f of. The parentheses do not impl multiplication. You can use letters other than f to name a function. The letters g, h, j, and k are often used to name functions. Matching Functions with Their Graphs Work with a partner. Match each function with its graph. ATTENDING TO PRECISION To be proficient in math, ou need to use clear definitions and state the meanings of the smbols ou use. a. f () = b. g() = + c. h() = 1 d. j() = A. B. C. D. Evaluating a Function Work with a partner. Consider the function f() = +. Locate the points (, f()) on the graph. Eplain how ou found each point. 5 f() = + a. ( 1, f( 1)) b. (, f()) c. (1, f(1)) d. (, f()) Communicate Your Answer. How can ou use function notation to represent a function? How are standard notation and function notation similar? How are the different? Standard Notation Function Notation = + 5 f() = + 5 Section. Function Notation 11

2 . Lesson Core Vocabular function notation, p. 1 Previous linear function quadrant What You Will Learn Use function notation to evaluate and interpret functions. Use function notation to solve and graph functions. Solve real-life problems using function notation. Using Function Notation to Evaluate and Interpret You know that a linear function can be written in the form = m + b. B naming a linear function f, ou can also write the function using function notation. f() = m + b Function notation The notation f() is another name for. If f is a function, and is in its domain, then f() represents the output of f corresponding to the input. You can use letters other than f to name a function, such as g or h. READING The notation f() is read as the value of f at or f of. It does not mean f times. Evaluating a Function Evaluate f() = + 7 when = and =. f() = + 7 Write the function. f() = + 7 f() = () + 7 Substitute for. f( ) = ( ) + 7 = Multipl. = = 1 Add. = 15 When =, f () = 1, and when =, f () = 15. Interpreting Function Notation Let f(t) be the outside temperature ( F) t hours after a.m. Eplain the meaning of each statement. a. f() = 58 b. f() = n c. f () < f(9) a. The initial value of the function is 58. So, the temperature at a.m. is 58 F. b. The output of f when t = is n. So, the temperature at noon ( hours after a.m.) is n F. c. The output of f when t = is less than the output of f when t = 9. So, the temperature at 9 a.m. ( hours after A.M.) is less than the temperature at p.m. (9 hours after a.m.). Monitoring Progress Help in English and Spanish at BigIdeasMath.com Evaluate the function when =,, and. 1. f() = 5. g() = 1. WHAT IF? In Eample, let f(t) be the outside temperature ( F) t hours after 9 a.m. Eplain the meaning of each statement. a. f() = 75 b. f(m) = 7 c. f() = f (9) d. f () > f() 1 Chapter Graphing Linear Functions

3 Using Function Notation to Solve and Graph Solving for the Independent Variable For h() = 5, find the value of for which h() = 7. h() = 5 Write the function. 7 = 5 Substitute 7 for h() Add 5 to each side. = Simplif. ( ) = Multipl each side b. = When =, h() = 7. Simplif. Graphing a Linear Function in Function Notation Graph f () = + 5. Step 1 Make an input-output table to find ordered pairs. 1 1 f () Step Plot the ordered pairs. Step Draw a line through the points. STUDY TIP The graph of = f() consists of the points (, f()). 8 f() = + 5 Monitoring Progress Help in English and Spanish at BigIdeasMath.com Find the value of so that the function has the given value.. f() = + 9; f() = 1 5. g() = 1 + ; g() = 1 Graph the linear function.. f() = 7. g() = + 8. h() = 1 Section. Function Notation 1

4 Solving Real-Life Problems Modeling with Mathematics Distance (miles) First Flight f() Hours The graph shows the number of miles a helicopter is from its destination after hours on its first flight. On its second flight, the helicopter travels 5 miles farther and increases its speed b 5 miles per hour. The function f() = 5 15 represents the second flight, where f() is the number of miles the helicopter is from its destination after hours. Which flight takes less time? Eplain. 1. Understand the Problem You are given a graph of the first flight and an equation of the second flight. You are asked to compare the flight times to determine which flight takes less time.. Make a Plan Graph the function that represents the second flight. Compare the graph to the graph of the first flight. The -value that corresponds to f() = represents the flight time.. Solve the Problem Graph f() = Step 1 Make an input-output table to find the ordered pairs. 1 f () Step Plot the ordered pairs. Step Draw a line through the points. Note that the function onl makes sense when and f() are positive. So, onl draw the line in the first quadrant. f() = From the graph of the first flight, ou can see that when f() =, =. From the graph of the second flight, ou can see that when f() =, is slightl less than. So, the second flight takes less time.. Look Back You can check that our answer is correct b finding the value of for which f() =. f() = 5 15 Write the function. = 5 15 Substitute for f(). 5 = 15 Subtract 5 from each side..8 = Divide each side b 15. So, the second flight takes.8 hours, which is less than. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 9. WHAT IF? Let f() = 5 75 represent the second flight, where f() is the number of miles the helicopter is from its destination after hours. Which flight takes less time? Eplain. 1 Chapter Graphing Linear Functions

5 . Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check 1. COMPLETE THE SENTENCE When ou write the function = + 1 as f() = + 1, ou are using.. REASONING Your height can be represented b a function h, where the input is our age. What does h(1) represent? Monitoring Progress and Modeling with Mathematics In Eercises 1, evaluate the function when =,, and 5. (See Eample 1.). f() = +. g() = 5. h() = + 9. r() = 7 7. p() = + 8. b() = v() = n() = INTERPRETING FUNCTION NOTATION Let c(t) be the number of customers in a restaurant t hours after 8 A.M. Eplain the meaning of each statement. (See Eample.) a. c() = b. c() = c(8) c. c(n) = 9 d. c(1) < c(1) 1. INTERPRETING FUNCTION NOTATION Let H() be the percent of U.S. households with Internet use ears after 198. Eplain the meaning of each statement. a. H() = 55 b. H() = k c. H(7) 1 d. H(17) + H(1) H(9) In Eercises 1 18, find the value of so that the function has the given value. (See Eample.) 1. h() = 7; h() = In Eercises 19 and, find the value of so that f() = f. 1. MODELING WITH MATHEMATICS The function C() = represents the cost (in dollars) of buing tickets to the orchestra with a $1 coupon. a. How much does it cost to bu five tickets? b. How man tickets can ou bu with $1?. MODELING WITH MATHEMATICS The function d(t) =,t represents the distance (in kilometers) that light travels in t seconds. a. How far does light travel in 15 seconds? b. How long does it take light to travel 1 million kilometers? f 1. t() = ; t() = 15. m() = + 15; m() = 7 1. k() = 1; k() = q() = 1 ; q() = 18. j() = 5 + 7; j() = 5 In Eercises 8, graph the linear function. (See Eample.). p() =. h() = 5 5. d() = 1. w() = g() = f () = Section. Function Notation 15

6 9. PROBLEM SOLVING The graph shows the percent p (in decimal form) of batter power remaining in a laptop computer after t hours of use. A tablet computer initiall has 75% of its batter power remaining and loses 1.5% per hour. Which computer s batter will last longer? Eplain. (See Eample 5.) Power remaining (decimal form) Laptop Batter p t Hours. HOW DO YOU SEE IT? The function = A() represents the attendance at a high school weeks after a flu outbreak. The graph of the function is shown. Number of students Attendance A() 5 5 A Week. PROBLEM SOLVING The function C() = represents the labor cost (in dollars) for Certified Hours Cost $1 Remodeling to build a deck, where $1 is the number of hours of labor. The table shows sample labor costs $19 from its main competitor, Master Remodeling. The deck is estimated to take 8 hours of labor. Which compan would ou hire? Eplain. 1. MAKING AN ARGUMENT Let P() be the number of people in the U.S. who own a cell phone ears after 199. Your friend sas that P( + 1) > P() for an because + 1 is alwas greater than. Is our friend correct? Eplain.. THOUGHT PROVOKING Let B(t) be our bank account balance after t das. Describe a situation in which B() < B() < B().. MATHEMATICAL CONNECTIONS Rewrite each geometr formula using function notation. Evaluate each function when r = 5 feet. Then eplain the meaning of the result. a. Diameter, d = r b. Area, A = πr c. Circumference, C = πr Maintaining Mathematical Proficienc Solve the inequalit. Graph the solution. (Section.5) a < 5 or a 1 > < k + <. d + 7 < 9 or d 1 > < 17. v or v r a. What happens to the school s attendance after the flu outbreak? b. Estimate A(1) and eplain its meaning. c. Use the graph to estimate the solution(s) of the equation A() =. Eplain the meaning of the solution(s). d. What was the least attendance? When did that occur? e. How man students do ou think are enrolled at this high school? Eplain our reasoning. 5. INTERPRETING FUNCTION NOTATION Let f be a function. Use each statement to find the coordinates of a point on the graph of f. a. f(5) is equal to 9. b. A solution of the equation f(n) = is 5.. REASONING Given a function f, tell whether the statement f(a + b) = f(a) + f(b) is true or false for all inputs a and b. If it is false, eplain wh. Reviewing what ou learned in previous grades and lessons 1 Chapter Graphing Linear Functions