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Mathematics

5.2 Notes

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5.2 ? Functions and Function Notation Consider the following set: Domain: Range: A function is like a machine, you put a number in (input or _________), the machine mashes it, stretches it, etc. and the machine spits out a new number (output or __________). This is a special type of relation because for every input there is only one output. (eg. If you put 3 into the machine, it will always output 6) ?Every function is a relation but not every relation is a function.? Example 1 then output first Multiply by 3 Add 2 input Equation Graph Ordered Pairs Table of Values Input Output -3 -1 0 1 3 Determining whether a relation is a function: {(1, 2), (0, 5), (3, 8), (-3, 2), (5, 9)}Example 2 b) -2 3 6c)

Relations and Functions Final Review

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3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 t (Min) 1 2 3 4 5 6 d (km) 1 2 3 4 5?1?2?3?4?5 x 1 2 3 4 5 ?1 ?2 ?3 ?4 ?5 y Relations and Functions Practice Questions 1. The following distance-time graph represents the distance (in kilometres) a person bicycled during a 50-min period. Describe a possible scenario. 2. The table of values shows the cost of movie tickets at a local theatre. Number of Tickets Cost ($) 1 12 2 24 3 36 4 48 (a) Is this a linear or non-linear relationship? Explain how you know. (b) Assign a variable to represent each quantity in the relation. Which variable is the dependent variable and which is the independent variable?

Chapter 11: Relations and Functions Test

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Name: _______________ Date: _______________Relations and Functions Practice Test Multiple Choice (1 mark each) Use the following information to answer the next question. A farmer read a study of soil fertility and mineral absorption. It contained the graph below. The graph illustrates the relative absorption of copper by the top and bottom of a wheat plant when the plant was subjected to various solutions containing a fixed amount of copper but having different concentrations of aluminum. From this graph, the farmer concluded that the relative absorption of copper consistently decreased at the A top of a wheat as the aluminum concentration increased B top of a wheat as the aluminum concentration decreased C bottom of a wheat as the aluminum concentration increased

AHSME 1992

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USA AIME 1992 1 Find the sum of all positive rational numbers that are less than 10 and that have denominator 30 when written in lowest terms. 2 A positive integer is called ascending if, in its decimal representation, there are at least two digits and each digit is less than any digit to its right. How many ascending positive integers are there? 3 A tennis player computes her win ratio by dividing the number of matches she has won by the total number of matches she has played. At the start of a weekend, her win ratio is exactly .500. During the weekend, she plays four matches, winning three and losing one. At the end of the weekend, her win ratio is greater than .503. What?s the largest number of matches she could?ve won before the weekend began?

AHSME 1991

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USA AIME 1991 1 Find x2 + y2 if x and y are positive integers such that xy + x+ y = 71x2y + xy2= 880. (0) Rectangle ABCD has sides AB of length 4 and CB of length 3. Divide AB into 168 congruent segments with points A = P0, P1, . . . , P168 = B, and divide CB into 168 congruent segments with points C = Q0, Q1, . . . , Q168 = B. For 1 ? k ? 167, draw the segments PkQk. Repeat this construction on the sides AD and CD, and then draw the diagonal AC. Find the sum of the lengths of the 335 parallel segments drawn. Expanding (1 + 0.2)1000 by the binomial theorem and doing no further manipulation gives ( 1000 0 ) (0.2)0 + ( 1000 1 ) (0.2)1 + ( 1000 2 ) (0.2)2 + ? ? ?+ ( 1000 1000 ) (0.2)1000 = A0 +A1 +A2 + ? ? ?+A1000, (0) where Ak = (1000 k )

AHSME 1990

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USA AIME 1990 1 The increasing sequence 2, 3, 5, 6, 7, 10, 11, . . . consists of all positive integers that are neither the square nor the cube of a positive integer. Find the 500th term of this sequence. 2 Find the value of (52 + 6 ? 43)3/2 ? (52? 6 ? 43)3/2. 3 Let P1 be a regular r-gon and P2 be a regular s-gon (r ? s ? 3) such that each interior angle of P1 is 5958 as large as each interior angle of P2. What?s the largest possible value of s? 4 Find the positive solution to 1 x2 ? 10x? 29 + 1 x2 ? 10x? 45 ? 2 x2 ? 10x? 69 = 0 5 Let n be the smallest positive integer that is a multiple of 75 and has exactly 75 positive integral divisors, including 1 and itself. Find n/75. 6 A biologist wants to calculate the number of fish in a lake. On May 1 she catches a random

AHSME 1989

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USA AIME 1989 1 Compute ? (31)(30)(29)(28) + 1. 2 Ten points are marked on a circle. How many distinct convex polygons of three or more sides can be drawn using some (or all) of the ten points as vertices? 3 Suppose n is a positive integer and d is a single digit in base 10. Find n if n 810 = 0.d25d25d25 . . . 4 If a < b < c < d < e are consecutive positive integers such that b + c + d is a perfect square and a+ b+ c+ d+ e is a perfect cube, what is the smallest possible value of c? 5 When a certain biased coin is flipped five times, the probability of getting heads exactly once is not equal to 0 and is the same as that of getting heads exactly twice. Let ij , in lowest terms, be the probability that the coin comes up heads in exactly 3 out of 5 flips. Find i+ j.

AHSME 1988

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USA AIME 1988 1 One commercially available ten-button lock may be opened by depressing ? in any order ? the correct five buttons. The sample shown below has {1, 2, 3, 6, 9} as its combination. Suppose that these locks are redesigned so that sets of as many as nine buttons or as few as one button could serve as combinations. How many additional combinations would this allow? 5 10 4 9 3 8 2 7 1 6 2 For any positive integer k, let f1(k) denote the square of the sum of the digits of k. For n ? 2, let fn(k) = f1(fn?1(k)). Find f1988(11). 3 Find (log2 x)2 if log2(log8 x) = log8(log2 x). 4 Suppose that |xi| < 1 for i = 1, 2, . . . , n. Suppose further that |x1|+ |x2|+ ? ? ?+ |xn| = 19 + |x1 + x2 + ? ? ?+ xn|. What is the smallest possible value of n?

AHSME 1987

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USA AIME 1987 1 An ordered pair (m,n) of non-negative integers is called ?simple? if the addition m+n in base 10 requires no carrying. Find the number of simple ordered pairs of non-negative integers that sum to 1492. 2 What is the largest possible distance between two points, one on the sphere of radius 19 with center (?2,?10, 5) and the other on the sphere of radius 87 with center (12, 8,?16)? 3 By a proper divisor of a natural number we mean a positive integral divisor other than 1 and the number itself. A natural number greater than 1 will be called ?nice? if it is equal to the product of its distinct proper divisors. What is the sum of the first ten nice numbers? 4 Find the area of the region enclosed by the graph of |x? 60|+ |y| = |x/4|.

AHSME 1986

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USA AIME 1986 1 What is the sum of the solutions to the equation 4 ? x = 12 7? 4 ? x ? 2 Evaluate the product ( ? 5 + ? 6 + ? 7)(? ? 5 + ? 6 + ? 7)( ? 5? ? 6 + ? 7)( ? 5 + ? 6? ? 7). 3 If tanx + tan y = 25 and cotx + cot y = 30, what is tan(x + y)? 4 Determine 3x4 + 2x5 if x1, x2, x3, x4, and x5 satisfy the system of equations below. 2x1 + x2 + x3 + x4 + x5 = 6 x1 + 2x2 + x3 + x4 + x5 = 12 x1 + x2 + 2x3 + x4 + x5 = 24 x1 + x2 + x3 + 2x4 + x5 = 48 x1 + x2 + x3 + x4 + 2x5 = 96 5 What is that largest positive integer n for which n3 + 100 is divisible by n + 10? 6 The pages of a book are numbered 1 through n. When the page numbers of the book were added, one of the page numbers was mistakenly added twice, resulting in an incorrect sum of

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