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Standard III -- Apply concepts related to functions First some general notes on functions. Background notes on Functions: Relation – a set of ordered pairs. Domain – the first number in an ordered pair. Range – the second number in an ordered pair. When stating domain and range, never repeat a number. Example {(2,3), (2,4), (7,4), (5,8), (3, 9)} Domain = {2,3,5,7} Range = {3,4,8,9} Notice the numbers are listed from least to greatest. Domain is the first number in an ordered pair and is considered the “x”. Range is the second number in an ordered pair and is considered the “y”. Think alphabetical order: Domain and Range; x and y D before R and x before y How to tell if a relation is a function. . . Remember the definition of Domain or “x” . . . When determining if a relation (a set of ordered pairs) is a function, it all depends on the answer to this question: Does a number a repeat in the domain or “x” position? If you say yes a number repeats, then the relation is NOT a function. If you say no a number does not repeat, then the relation IS a function. {(1,2), (2,3), (4,5)} Domain is {1,2,4} did you repeat a number – NO, then it IS a function. Find the range given domain? When a question has the phrase, find the range given the domain it is a simply PII problem (Plug It In). Example: Find the range of y = 2x + 4 if domain is {1,2,4}. Since there are three numbers given in the domain, that means there are three answers. Write the equation three times. Then PII the domain… see next slide. Solve y = 2x + 4 y = 2 x + 4 domain is {1,2,4} y=2(1)+4 » y=2+4 y=2(2)+4 » y=4+4 y =2 ( 4 ) + 4 » y = 8 + 4 » y=6 » y =8 » y = 12 Y remember represents range so you are basically solving for “y”. The final answer is y or range is {6,8, 12} A Quick way to determine if a graph is a function… If you have a U or a V that opens up or down, then the graph IS a function. If you have a U or a V that opens sideways, then it is NOT a function. Horizontal Lines, then it IS a function. Vertical Lines, then it is NOT a function. All other shapes????? Other shapes on graphs can be determined if it is or is not a function by using the vertical line test. A vertical line is drawn on the shape. The vertical line can only touch or cut the shape once and only once to be a function. Not a function IS a function Mappings????? A mapping is usually two vertical boxes or two ovals with numbers inside of each. This is another method to list a relation (set of ordered pairs). The first box or oval represents the x or domain. The second box or oval represents the y or range. Now remember your definition of a function – x or the first number cannot repeat…so only ONE LINE can come from each number in the X column. Meaning each number in the X column can be matched with ONLY one number in the Y column. Standard 3 – Objective 1 Answer is A. Remember that a graph with a U or a V opening up, is ALWAYS a function. Choice B is a U opening sideways which is NEVER a function. Choice C and D are an oval and circle which are NEVER a function. 2. Which of these mappings is not a function? Answer: C Look at column X, notice that the number 2 is mapped to a 2 and a 4 in column f(x). F(x) is another way to write y. Which of these represents the data in the table? Answer is B. This is a PII problem (plug it in). Start with equation A. Plug in the first x value into the equation and perform Order of Operation. The answer should match the first y value. Remember for an equation to be the correct one, each of the x and y values must check for the equation. Answer is D This is a PII problem. Notice when you plug in 0 into the x and solve the problem the y or the answer is -5. That means the answer is either C or D. After checking all the values in C and D, choice D is the only choice which all three x values matches all three y values. Answer is A. The best way to solve this type of problem is to write the mapping as ordered pairs. (0,1), (1,2), (2,3), (3,4) Now plug the x value into the equation and solve for y. The only equation that checks for every ordered pair is A. Answer is D. This is a PII problem as well. Plug in the first x value into the x into the equation. Remember to take the absolute value first before adding 1. The answer you find should match the first y value. Remember every value MUST check. Answer is C. Choice C is the only relation that the X value does NOT repeat in the x position. Answer is B. Choice B is a V that opens down. Answer is B. Remember a vertical line is NEVER a function. Standard 3 Objective 2 Answer is C. Range is the second number in an ordered pair. Underline the second number in each ordered pair and write the number on your paper. Remember do not write a number more than ONCE. Answers are listed from least to greatest. Answer is D. This is a plug it in problem. Plug in the first value for X and then apply order of operations to the problem. y = 3x² - 5 {-2,0,1} y = 3(-2)² - 5 y = 3(4) – 5 y = 12 – 5 y=7 NOTICE ONLY CHOICE B HAS A 7 IN THE ANSWER. Answer is A not D. If you calculated answer D then you have a common mistake. Look closely at the problem. Did you notice there is a negative sign in front of the x²? This is really -1 x². Remember order of operations you must do the exponents before multiplying. Therefore, you first must square the number then multiply by -1. Look at the next slide for the step by step procedure for solving this problem. f(x) = -x² + 2x – 3 , what is f(4) The f(4) is the plug in value so you will plug in 4 for x. f(x) = -x² + 2x – 3 = = = = = -1(4)² + 2(4) – 3 -1 (16) + 8 – 3 -16 + 8 – 3 -8 – 3 -11 Answer is D. Notice the problem is stating RANGE which is the y-axis not the x-axis which most students mistakenly read. Draw a line from each dot to the y-axis and read the number on the y-axis from least to greatest. Also notice the dots are solid, therefore the symbols for less than should have a line under each one.