Lesson #17 Function Introduction

Transcription

1 Lesson #17 Function Introduction A.A.37 A.A.40 A.A.41 Define a relation and function Write functions in functional notation Use functional notation to evaluate functions for given values in the domain 1. A relation is any set of ordered pairs with an input (first item) and an output (second item). The idea of a relation is similar to the word relationship where one thing has an impact on another.. The domain of a relation is all of the possible inputs or -values. 3. The range of a relation is all of the possible outputs or y-values, also known as the function s values. 4. A function is a type of relation where each input produces one and only one predictable output. a. The mathematical concept of a function epresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary or the starting point (the independent variable, argument of the function, or its "input") and the other as secondary or the result (the value of the function, or "output").... b. In simple terms: i. You start with something. (the -value or input value) ii. You do something to it (the function) iii. You get something else as a result: (the y-value, output value, or function s value) Eamples: When you work for hours you get y dollars. The amount of money you make (y) can be thought of as a function or result of the amount of hours you work (). When the temperature is above 3 degrees, ice will melt. The state of the water is a function or result of the current temperature. First, remember this: While parentheses have, up until now, always indicated multiplication, parentheses do not Function Notation: y=f() indicate multiplication in function notation. "f()" means "plug in a value for "; it does not mean "multiply f and "!! If E) f()=+3 you read "f()" as "f times ", I can guarantee that you'll drive your instructor batty! (By the way, that is not a Translation: suggestion!) - Purplemath.com ~ 1 ~

2 1. The amount of money you make each week depends on how many hours you work at $8.00 per hour plus your weekly allowance of $5. The maimum number of hours you can work is 40. f() = 8+5 Input Work (Function rule) Output Ordered Pair In Function Notation Overall Domain: Overall Range:. The cost of a limo ticket depends on how many people are in the limo when the limo costs a total of $500. The maimum number of people is 10. You can name a function any 500 letter. You can also use any C( p) p variable instead of. Input Output Ordered Pair In Function notation Overall Domain: Overall Range: 3. The difference between the cost of your $30 pants and any other item in the store. d( ) 30 In Function Notation Ordered Pair Overall Domain: Overall Range: ~ ~

3 The area of a circle depends on the radius: A() r r In Function Notation Ordered Pair Overall Domain: Overall Range: If f ( ) 7, find f(4). If ga ( ) 7a, find g(-4). a 3 m If rm ( ) 3 7, find r(). If f ( ) 6, find f(4). Other representations of Functions E) f() = 8+5 Given Ordered Pairs Mapping Diagrams Graphs ~ 3 ~

4 Summary: Why use f() instead of y=? 1. You used to say "y = + 3; solve for y when = 4". Now you say "f() = + 3; find f(-4)." This is much more stream lined.. When you found y, you could no longer see the -value. Now you can write f(-4)= -5 and see both the input and the output in a concise form. 3. Now you can name functions. Instead of each function being y=, you can have names like f(), g(), h(), s(t), etc. The point is that you can use more than one function at a time, and not mi them up, wondering "Okay, which 'y' is this, anyway?" 4. The notation can be eplanatory: "A(r) = (π)r " indicates the area of a circle, while " h( t) 16t 8t " could indicate the height of an object t seconds after it was thrown. Copyright Elizabeth "f()" gives you more fleibility and more information. There are many more reasons for using function notation that will be given later in the unit! ~ 4 ~

5 Lesson #18 Relations, functions, and 1-to-1 functions Sometimes you don t know what you re going to get A.A.38 A.A.43 A.A.5 Determine when a relation is a function Determine if a function is one-to-one, onto, or both Identify relations and functions, using graphs 1. f(s): input a students name and the output will be his or her biological father Input Output Ordered Pair In Function notation. p(s): input a students name and the output will be his or her parent Input Output Ordered Pair In Function notation What is different about the second relationship? Relations Functions Relation: any set of ordered pairs One-to- One Functions Function: a relation where each input produces only one output ~ 5 ~

6 A function is a well-behaved relation. Just as with members of your own family, some members of the relation family are better behaved than other. (This means that, while all functions are relations, since they pair information, not all relations are functions. Functions are a sub-classification of relations.) When we say that a function is "a well-behaved relation", we mean that, given a starting point, we know eactly where to go. That is, given an, we get eactly one y. purplemath.com On a Graph General Tests to See if Relations are Functions A Relation that is a Function A Relation that isn t a Function With Ordered Pairs {(1,), (,4), (3,5)} {(1,), (, 4), (3, 5), (, 6), (1, -3)} You can also graph points or the equation of a relation and use the vertical line test. With an equation y y 1. Which set of ordered pairs is not a function? (1) {(3,1), (,1), (1,), (3,)} () {(4,1), (5,1), (6,1), (7,1)} (3) {(1,), (3,4), (4,5), (5,6)} (4) {(0,0), (1,1), (,), (3,3)}. Which two sets of ordered pairs represent a function? (1) {(3,-), (-,3), (4,-1), (-1,4)} () {(3,-), (3,-4), (4,-1), (4,-3)} (3) {(3,-), (1,-3), (5,-4), (1,-5)} (4) {(3,-), (5,-), (4,-), (-1,-)} ~ 6 ~

7 3. Determine if the graph of each relation is a function (a-f). 4. Decide if each mapping diagram is a function. a) b) ~ 7 ~

8 Equations: situations where you cannot solve for one unique y. A. E) =6 B. E) y 9 C. E) 3y 1 Which of the following are functions? If it is a function, write it in function notation. a) y 10 4 b) 15 c) y d) y 3 6 e) y = 6 f) y 3 g) = -15y - 11 h) y = i) y - = -40 j) 86 = 6y + 11 One-To-One Functions One-to-One Function: a FUNCTION where each output value ALSO comes from only one input value. NOTE: For something to be a 1-to-1 function, you must determine that it is a function first. Method 1-to-1 Function Function, not 1-to-1 Graph Ordered Pairs {(,3), (6,8), (3,5), (7,4)} {(3,), (5,), (6,3), (4,)} ~ 8 ~

9 Equation (it is good to know what types of equations will be one to one functions, but you will usually use a graph to decide). y=3- y 3 Go back through problems 1 to 4 on pages 6 and 7. If you decided that problem was a function, decide if it is 1-to-1. Why does the vertical line test work to see if a relation is a function? Why does the horizontal line test work to see if a function is 1-to-1? ~ 9 ~

10 Lesson #19 Domain, Range, and Pictures of Functions A.A.51 Determine the domain and range of a function from its graph If p( ) 4, find p(-7). What does it mean for something to be continuous? Finding the Domain and Range from the Graph of a Relation We can draw a picture of a function using a graph. The -values are the inputs and the y-values (function values) are the outputs. The domain and range can be epressed in roster notation if there are a finite set of elements. Simple relations with finite domain and range Find the domain and range for each relation. Afterwards, decide if the relation is a function. 1) {(3,11), (,7), (1,3),(0,-1)} ) {(1,4), (1,), (4,6)} 3) ) ~ 10 ~

11 5) Continuous Relations If the relation is continuous, the domain and range can be epressed in either set or interval notation. (Assume questions 3, 4, 5, and 6 have arrows) ~ 11 ~

12 Find the domain and range for each of the following functions. F() G() H() Sometimes you are given a set of domain values and asked to find the range for that set of values only. For the following domains, find the range of F() from the graph above. Domain of F() 1,4 Range of F() 6, 4,0,1 ~ 1 ~

13 Lesson #0 - Parent Functions & Restricted Domains A.A.39 Determine the domain and range of a function from its equation Parent Function: A basic function with the essential qualities of a particular type. Parent functions are used as a starting point for more complicated functions of that type. Basically, all other graphs of each type are children or transformations of a parent function. We will do more with transformations net unit. Each of the mathematical operations we have worked with this year can also be thought of as a function with input and output. Name General form Parent Graph 1 to 1? Domain Range Other Eamples Function Linear y=m+b y = f() = Quadratic y a b c y g( ) Absolute Value y a h k q() ~ 13 ~

14 Name General Form Equation Graph 1 to 1? Domain Range Other Eamples Square Root y a h k j( ) 3 Cubic y a b c d c( ) 3 Rational 1 y a k h r ( ) 1 ~ 14 ~

15 Restricted Domains (There are some input () values that will not work) A. For most functions, you can use any real number (ALL REALS) as an input. For eample, you can square any real number, take the absolute value of any real number, and multiply any real number by another. Use that information to find the domain for each of the following functions: g( ) 6 7 a. b. c. f ( ) 4 7 m( ) 5 3 B. For rational functions you cannot use any real number because the. r ( ) Summary 3 6 E) f( ) 3 Cannot have zero in the denominator. Find: f(0)= f(1)= f()= Therefore, the domain of f() is ALL REAL NUMBERS EXCEPT. Restriction: ) Domain: R, ) ~ 15 ~

16 C. For square root functions you cannot use any real number because the number in the radical cannot be. E) f ( ) 7 Find: f(6)= f(7)= f(8)= 1 Therefore, the domain of f() is ALL REAL NUMBERS GREATER THAN OR EQUAL TO 7. Summary s( ) 8 y 4 y 4 11 y 10 3 Can only take the square root of a positive number. Restriction: Domain: [4, ) ~ 16 ~

17 Lesson #1 Direct and Inverse Variation A.A.5 Use direct and inverse variation to solve for unknown values Direct and Inverse Variation are two types of functions where the input and output are proportionally related to each other. The input and output vary directly when they increase or decrease in proportion to each other. The input and output vary inversely when the input increases in proportion to a decrease in the output. Look at each of the following situations and decide if the input and output are related directly or inversely. Input Output Relationship Number of DVDs bought Total cost of the DVD s Cost of a bracelet How many you can buy with $40 Speed Speed Number of people at a birthday party Daily caloric need Number of students in a class Number of students in a class Distance travelled in 3 hours Time to get to your destination Amount of the birthday cake for each person Amount of food you can eat Time the teacher can help each student individually Noise level if each person is talking For this course, you will need to solve word problems that involve inverse and direct variation. Usually, the problem will tell you which one to use. You just have to know how to set up the equation to solve. There are a couple of key phrases you will be looking for. In the problem, instead of direct variation, you might see varies directly or directly proportional. When you see this wording you should set up a normal (direct) proportion to solve: Direct Variation: y 1 y 1 Instead of inverse variation, you might see varies inversely or inversely proportional in a word problem. Sometimes the word indirect is used in the place of inverse. For the directly proportional problem, the operation that is taking place between the two variables is division. ~ 17 ~

18 The inverse of division is. Therefore the setup for inverse variation problem is: Inverse Variation: 1 y1 y We will use situations from the table on the previous page for the first couple of eamples. 1) The number of DVDs is directly proportional with the total cost of the DVDs. If 5 DVD s cost $40, how much would 8 DVD s cost? ) The cost of a bracelet is inversely proportional to the number of bracelets you can afford to buy with a certain amount of money. If the bracelets cost $5, you can buy 10 of them. How many bracelets could you buy if they cost $5? 3) Your speed varies directly with the distance you travel in a set amount of time. If you drive 60mph, you can travel 500 miles. In the same time period, how far could you travel going 55mph. Round to the nearest mile. 4) The number of students in the class varies inversely with the number of minutes the teacher can spend helping each student individually. In a class of 15 students, the teacher can spend.9 minutes with each student. How many minutes can the teacher spend with each student in a class of 5? Round to the nearest tenth of a minute. 5) If y varies inversely with, and is 16 when y is, what is y when is 8? 6) If and y are inversely proportional, and is 7 when y is 3, what is when y is 9? ~ 18 ~

19 7) If and y vary inversely and = - when y = 10, what is when y = 15? 8) If and y are directly proportional and =.5 when y = 4, what is y when = 4? 9) The number of typists is inversely proportional to the time it takes to finish a job. If 4 typists can complete the typing of a manuscript in 9 days, how long will 1 typists take to complete the manuscript? 10) If a man can drive from his home to Albany in 6 hours at 45 miles per hour, how long will the trip take if he drives at 60 miles per hour? How much time will be saved at the higher speed? 11) The number of gallons of gasoline used by a moped varies directly as the number of miles it travels. If a moped uses gallons of gasoline to travel 115 miles, how many gallons is needed to travel 64.5 miles? ~ 19 ~

20 A.A.40 A.A.41 Lesson # Function Notation Write functions in functional notation Use functional notation to evaluate functions for given values in the domain f()=-3. NOT: What does f() mean? The epression equal to f() is also known as the rule for f(). For eample, the rule for f() is -3 in the eample to the left. a) f ( ) 4, find f (3). b) g( ) 3 4, find g ( 1). c) f ( ) 4, find f( m ). d) h( m) 4m 15, find h ( 3). Substitutions: Steps Take out all of the s. Plug in the number, variable, or epression that takes the place of in the blanks. Evaluate or simplify the epression e) f ( ) 3, find 3f(). f) h ( ) 5 3, find h. g) Given that f() = 3 +, find f ( - h) + k. ~ 0 ~

21 Reading function values off a graph The function, g() and f() are graphed. f(): g(): a) Find g(-3) b) Find f(1) c) Find g(0) d) Find g(3) + f(-). e) Estimate the value(s) of where f() = -1. f) Estimate the value(s) of where g() = 0. Other Operations and Equations with Functions Just as you can add, subtract, multiply, and divide numbers or variables, you can do the same with functions. If f()=4+3 and g( ) 3, complete problems a-d. a) f()+g()= b) When does f()=g()? c) For what value(s) of does f() = -5? KEY IDEA: When you still see the name of the function such as f(), the is still there. Therefore, you do not plug in anything for. The function stays the same. d) Write the rule for f(+1)-7. ~ 1 ~

22 If f()= and g( ) 7 e) f() - g()=, complete problems e-h. f) f()g()= g) Write the rule for f(7y-4). h) When does g(+)=10? i) The revenue, R(), from selling units of a product is represented by the equation R ( ) 35, while the total cost, C(), of making units of the product is represented by the equation C ( ) The total profit, P(), is represented by the equation P ( ) R( ) C( ). For the values of R() and C() given above, what is P()? (1) 15 (3) () (4) j) If f( ) and g( ), find f( ) g ( ) for all values of for which the epression is defined and epress your answer in simplest form. ~ ~

23 Lesson #3 Compositions: Using Functions in a Row A.A.4 Find the composition of functions Sometimes one function or operation is not enough to get the job done. When two or more functions are used in a row to get a final answer it is called a composition of functions. The output from the first function is the input for the second function. Notation: f ( g( a)) ( f g)( a) f g( a) where a is the initial input value. ALWAYS rewrite a composition in because this helps us see the order of operations better. Compositions from ordered pairs f() = {(-,3), (-1,1), (0,0),(1,-1),(,-3)} g() = {(-3,1), (-1,-), (0,), (,), (3,1)} 1) f(1)= ) g(-1)= 3) g(f(1))= 4) ( f g )(0) = 5) g f ( 1) = 6) f()=3-. g()=4-5. a. Find g(f(3)). Compositions from equations Steps for Composition Evaluations If necessary, rewrite in the form that is easier to read, g(f(a)). Find f(a), the function value in the inner parenthesis. b. Find ( f g )( 1). (Rewrite as ). Plug that value into the function in the outside parenthesis, in this case g. Evaluate and write a concluding statement. E) g(f(3))= ~ 3 ~

24 7) Suppose f and g 8. Find g(f(1)). 8) If f 1 1 g a 5 a, Find g f. 9) The temperature generated by an electrical circuit is represented by t f ( m) 0.3m, where m is the number of moving parts. The resistance of the same circuit is represented by r g( t) 150 5t, where t is the temperature. What is the resistance in a circuit that has four moving parts? (1) 51 (3) 174 () 156 (4) 8,670 Compositions from Graphs f(): g(): 10) g(-1)= 11) f(1)= 1) ( f g )( 1) = 13) ( g f )( 3) = ~ 4 ~

25 Writing a New All-in-One Function Instead of repeatedly using two functions in a row, you can make up a new function or rule that is the composition of the two functions. Think of this like a machine that will wash your clothes first (f()) and then put them in the dryer (g()) to get a finished product. This all in one machine would be g(f()) because it does f first then g. Different Notations: f ( g( )) ( f g)( ) f g( ) All read: The following eample is worked out for you with step by step instructions. Given f() = + 3 and g() = + 5, find (g o f )(). If necessary, rewrite the composition in the form that is easiest to read: f ( g( )). (g o f )() = g (f()) Substitute the inside function in for its name in the inner parenthesis. Substitute the epression for in the function named outside the parenthesis. SIMPLIFY the resulting function and write a concluding statement. = g ( + 3) = ( ) setting up to insert the input = ( + 3) + 5 = ( ) + 5 = = Answer: f(g())= Eample: If f ( ) 7 and g( ), find f g(). Relation to real life: if two processes, machines, or functions typically work one after the other, it isn t long before someone invents an machine to make life easier. You now know that all-in-one functions where two functions that occur in a sequence are put together are called. THERE IS NOTHING TO SOLVE. YOU ARE JUST WRITING A NEW FUNCTION! ~ 5 ~

26 1) If f()=3- and g()=4-5, a. Find g(f()). b. Find ( f g)( ). ) If f 7 and g, give the rule for g f 3) f 3 and g 5. a. What is the rule for f g? b. What is the rule for g(f())? 4) f 1 1 and g a 5 a find the rule for f g ( a). ~ 6 ~

27 Lesson #4 Inverse of a Function A.A.44 Define the inverse of a function Given a function f(), the inverse notation is f 1 ( ). Basic Function Rule Evaluate the function when the input is 5 Inverse Function Evaluate the inverse function using the output from column as the new input Add four to the number: f()=+4 Multiply the number by 3 g()=3 Take the square root of the number h( ) We can see that finding the inverse does not mean you always change the signs, flip the fractions, or any other specific process because the inverse is dependent on the type of operation(s) in the original function. We need a method that will work in ALL SITUATIONS including two step linear equations, points on a graph, or other more difficult functions we will encounter in the future. Finding the inverse: In the original function, we plug the input,, into the function to get the output,. Since the inverse undoes the work of the original function, the output will map back to the input. In other words, the and change places. If you have function notation, f(), you can replace it with y to perform this operation. Therefore, to find the inverse of any function,. Use this method to find the inverses f 1 ( ), g 1 ( ), and h 1 ( ) from the table above. ~ 7 ~

28 Function A: {(3,11), (,7), (1,3),(0,-1)} Domain: Range: With Ordered Pairs Method: Switch the and the y. 1 A : Domain: Range: What happened to the domain and range? Find the inverse of Function B: {(1,4), (3,), (4,6)} Graphing Linear Equations Reminder: y=m+b m=slope b=y-intercept - Start from b on the y-ais. up( ) or down(-) - Count using m. over Try this with the 1 st eample before finding the inverse. Finding the Inverse of a Function Epressed as an Equation E) g()=3-4 Method: 1) If necessary, substitute y for f(). ) Switch the and the y. 3) Solve for y. 4) If necessary, substitute f 1 ( ) back in for the new y. E) f()=5+. ~ 8 ~

29 E) g()= 3 E) f()= 1 4 Inverses and graphing Method: Switch the and the y values of specific points. ~ 9 ~

30 Lesson #5 Inverses, Functions, & Compositions A.A.45 Determine the inverse of a function and use composition to justify the result f()=3+7 Review: g()= 5 Using the functions above, find: a) g(-1) b) [f(-4)+] c) g()f() d) f(g()) e) g(f(3)) f) When g()=0 g) f 1 ( ) ~ 30 ~

31 You will recall that a function is a relation where each input produces only one output. We use the to see if a graph is a function. A one-to-one function is a function where each output ALSO comes from only one input. Each member of the domain and range only appears once (one-to-one). We use the to see if a graph is a one to-one function. Do we ever perform the horizontal line test if the vertical line test does not work? Why or why not? When we find the inverse of a function, what happens to the and y values? Therefore, if a function is, the inverse will also be a function. Is the inverse a function? Eample: Function: {(3,), (5,), (6,3), (4,)} Inverse: Is the inverse a function? For each of the problems below decide if the inverse is a function. Justify your answer. a. y 5 9 b. y 9 c. {(3,4), (5,4), (,1)} d. {(3,7), (5,4), (,1)} e. f. ~ 31 ~

32 Compositions and Inverses Let s think of walking forward 5 steps as a function. Function: Walking Forward Five Steps Inverse Function: Walking Backward Five Steps B. Function: Standing in the Doorway + 5 steps When you perform the function and then its inverse, your final output is always the original input. From two lessons ago, we know that performing two functions in a row is called a composition of functions. A. Input: Standing in the Doorway E. (Final output) D. Inverse Function: C. Output: 5 steps inside classroom 5 steps inside classroom - 5 steps Function: f ( ) Inverse: f ( ) B. Function: (3)+5 No matter what number you plug into the function, you will get the same number back if you plug your output into the inverse. Therefore, if you plug in the variable, you will get back. A. Input: 3 E. (Final output) D. Inverse Function: (11-5)/ C. Output: 11 f 1 ( f()) 1 f ( f ( )) 1 f ( f ( )) The original input and the final output are the same. By doing the function and then its inverse (or visa-versa) you will always end up where you started. ~ 3 ~

33 Theorem: The value of the composition of a function and its inverse will always be f ( g( )) the original input. Given f() and g(), if you can show that, you will g( f ( )) have proven that f() and g() are inverses of each other. 1. f ( ) 5 and g( ) 5. Are f() and g() inverses of each other? Hint: Does f ( g( ))? Does g( f ( ))?. 1 f ( ) 6 and g( ) Are f() and g() inverses of each other? 3 Hint: Does f ( g( ))? Does g( f ( ))? 3. f ( ) 8 a. find f 1 ( ) b. find f 1 ( f (5)) c. find f 1 ( f ( )) d. find 1 f ( f ( )) ~ 33 ~