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~ FUNCTIONS ~ ( MULTI-VALUED v.s. SINGLE-VALUED FUNCTIONS) I like to think of functions a little differently than most teachers because I believe it helps students understand: A function is single-valued if, for every x, there is just one associated y-value. (Remember, the y-coordinate is called the „value‟). The Big 8 are all single-valued functions, or for short, functions. A function is multi-valued if there are two or more values of y for a single x-coordinate. An example of this is a circle. The word „FUNCTION‟ by itself means SINGLE-VALUED. KEY POINT: Numbers can be divided up into two groups, positive and negative. If you say the number 5, you don‟t have to say „positive 5‟ – you just say „five‟. Similarly, when the word "function" is used without the words 'single-valued' or 'multivalued' in front of it, it is understood to mean single-valued. Unless otherwise stated, all functions in this book will be single valued. If a function is multi-valued, it will be called multi-valued. So in our class, there will be „functions‟ and „muti-valued‟ functions. Single-valued functions (get used to calling them functions) are generally easier to work with. In a later chapter we perform 'operations' on (single-valued) functions because operations on multi-valued functions can complicate things. Multi-valued functions can arise naturally in the chapter on inverse functions as inverses of single-valued functions. By appropriate piecewise domain restrictions on the original functions, inverses can be turned into (single-valued) functions. For example f(x)=x2 is single-valued but f-1(x) is multi-valued. If f(x)=x2 is restricted to x 0 , then f-1(x) is singlevalued. In fact f-1(x)= x . More on this later. FUNCTIONS 1 REPRESENTING FUNCTIONS A function can be represented algebraically as in y = x2. A function can be represented using function notation as in f(x) = x2. A function can be represented in a table as: x y -2 4 -1 1 0 0 1 1 2 4 3 9 Etc. A function can be represented by a set of ordered pairs of numbers, as {(-2,4), (-1,1), (0,0), ), (1,1), (2,4), (3,9)} A function can be represented by a graph, as in (draw a graph of y = x2 here). A function can be represented verbally as: “the function y equals x squared”. Which form(s) you choose will depend on a variety of things including the nature of the function, what it is you are trying to do, your personality, what tools you have available, what you are most comfortable with, the numbers involved, your familiarity with the function, your grasp of the situation. FUNCTIONS 2 SINGLE-VALUED FUNCTIONS The Vertical Line Test for (single-valued) function. It is easy to tell whether a function is multi-valued or single-valued by looking at its picture form. This is called the vertical line test for "function". The vertical line test says that if you pass a vertical line across a graph and if it ever touches more than one point at a time, it is multi-valued. Otherwise it is single-valued. Remember that if just the word 'function' is used by itself, with no qualifier, it is singlevalued. As examples, consider again a circle (multi-valued) and a straight line (single-valued). We study mostly single-valued functions in this book. It is also easy to tell whether a function is single-valued or multi-valued by looking at the table form. If a table entry in the x column appears more than once with a different y-value for each (as in the circle above where x=6), then it is multi-valued. If each x appears just once, then it is single-valued, and hence called a function. FUNCTIONS 3 MULTI-VALUED FUNCTIONS Example: A Circle A function is multi-valued if there are two or more values of y for a single x-coordinate. An example of this is a circle. Consider the circle of radius 10 centered at the origin below. It is multi-valued because for x = 6 there are associated multiple values of y, namely 8 & -8. Hence it is multi-valued function. All ordered pairs (x,y) such that x2 + y2 = 100 Algebraic: x2 + y2 = 100 Picture: (draw a circle here) Table: x y 0 10 6 8 6 -8 Etc Read p193; p198#1-17odd Circles Pop Quiz: Find the equation of a circle: 1) If (-4,-1) & (4,1) are endpoints of a diameter. Ans: x2 + y2 = 17 2) With center (3,-2) and solution point (-1,1). Ans: (x-3)2 + (y+2)2 = 25 = 52 Generally, an equation with y2 or |y| will be multivalued. Determine if the following represent single-valued or multi-valued functions: y = x2 y = |x| x2 + y = 25 y=x 2 2 x =y |x| = y x + y = 25 y-x=0 2 2 2 y =x |y| = x x + y = 25 x-y=0 2 x=y x = |y| x=y FUNCTIONS 4 PIECEWISE FUNCTIONS Read page 252 Pop Quiz: f(x) = x if x > 0 and x - 1 if x ≤ 0. Find f(0), f(1), and f(-1). FUNCTIONS 5 ZEROS OF FUNCTIONS Zeros - Deeper Thinking Tools to find the zeros of f(x) are the values of x where f(x) = 0: factor, square root property, graph and find x-intercepts. These are equivalent: f(x) = g(x) f(x) - g(x) = 0 FUNCTIONS 6 FINDING FORMS OF A FUNCTION You can easily generate a table of values from an equation of the function. For example, the table for the function f(x) = x2 is: x y 1 1 2 4 -3 9 Etc. It can be a little more difficult to find the algebraic form from a table. In the example above, you can see that each y value is the square of each x number, so the equation is y = x2, or equivalently f(x) = x2. That was not hard. Try these: Match the data with one of the following functions: y = 1 x x x x2 3 x2 x x x y x y [[x]] x2-3 -x x y x y x y 4 0.25 4 4 1 1 1 1 4 16 1 1 0 0 4 4 1 1 0 0 4 1 0 4 1 0 0 undef 0 1 4 1 4 1 4 1 4 1 4 1 4 1 0.25 1 4 x y x y x y x 4 1 0 4 13 1 2 0 3 1 4 1 4 x 1 16 y 0 0 4 2 1 1 0 0 1 4 1 2 1 4 4 undef 1 undef 1 2 0 0 4 1 0 1 4 1 48 1 4 4 48 1 3 Now try these - match the data with one of the following functions: c f(x)=cx; g(x)=cx2; h(x)=c x ; r(x)= . x Table 1 Table 2 Table 3 x y x y x y 4 1 4 8 4 32 1 1 1 32 1 2 4 0 0 0 Undef . 0 0 1 1 32 1 2 1 4 4 8 4 32 4 1 undef 1 4 y 2 13 Table 4 x y 4 6 1 3 0 0 1 4 3 6 These are also good frontloading for the abstract thinking required for the c1f(c2-c3) concepts of transformations. FUNCTIONS 7 DOMAIN & RANGE Review the domain and range of the 8 basic functions from Thinking 3 by: 1) Use the Parametric mode of TI-8x to demo Domain and Range! 2) Issue a pop quiz: Pop quiz: Find the domain and range of the 8 basic functions - make a table. (See Thinking 3 for answers): f(x) D R choices: (-∞,0); (-∞,∞); (0,∞); (-∞,0]; [0,∞); (-∞,0) U (0,∞) 3) Pick two students to point their finger on the x-axis (domain) or y-axis (range) as I move mine along the picture of the function. f(x)=x2 works well for this. FUNCTIONS 8 SUMMARY OF KEY POINTS 1) "value" of a function (the y-coordinate) 2) single-valued v.s. multi-valued 3) „function‟ means single-valued 4) domain v.s. range; 5) domain & range of the Big8 6) zeros, x-intercepts, roots 7) forms: graphic, algebraic or y=, table, verbal, function or f(x) or f(x)=(something), a set of ordered pairs {(x,y)}.