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Function

Chapter 11: Relations and Functions Test

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Algebra
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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 1988

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Algebra
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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 1984

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Algebra
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American Mathematics Competitions
Function
Area
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USA AIME 1984 1 Find the value of a2 + a4 + a6 + ? ? ?+ a98 if a1, a2, a3, . . . is an arithmetic progression with common difference 1, and a1 + a2 + a3 + ? ? ?+ a98 = 137. 2 The integer n is the smallest positive multiple of 15 such that every digit of n is either 8 or 0. Compute n15 . 3 A point P is chosen in the interior of 4ABC so that when lines are drawn through P parallel to the sides of 4ABC, the resulting smaller triangles, t1, t2, and t3 in the figure, have areas 4, 9, and 49, respectively. Find the area of 4ABC. A B C t3 t2t1 4 Let S be a list of positive integers - not necessarily distinct - in which the number 68 appears. The average (arithmetic mean) of the numbers in S is 56. However, if 68 is removed, the

Inverse Functions

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Algebra
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Inverse functions
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Inverse Functions Given a function , if there is a function such that which equals the identity function The function is said to be "invertible" "undoes" what "does", and vice versa Such a function is called the inverse, denoted as (the inverse of ) The notation is not to be confused with an exponent In some cases the inverse of a function can be found through algebraic methods CONSIDER: Given to determine we must find a functions that must undo But, recall the set of outputs from , to undo we take Thus Observe that: Not every function has an inverse CONSIDER: But is not a function. An inverse only exists when different inputs in the domain always yield different outputs in the range Such functions are called one-to-one

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