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Unit 2 Section 1 and Section 2 - UNIT II A Theory of...

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UNIT II. A Theory of Relations and Functions In the previous unit we discussed properties of numbers, and how the properties are used to evaluate expressions and solve equations. In this unit we will concentrate on equations that have two variables. These equations tell how one variable is related to the other. Consider the equation A= πr 2 , where A is the area of the circle and r is the radius. For any r, we can find a value for A, say if r = 1then A = π, if r=2 then A = 4π and so on depending on the value of the radius. OBJECTIVES: At the end of this unit, you must be able to: 1. define relations, functions and inverse functions 2. state the domain, range, intercepts, and symmetry of the functions and relations 3. differentiate relations from functions 4. perform operations on functions and 5. sketch the graphs of functions and their inverses. 2.1 Relations and Graphs Recall the definition of the cross product: Let A and B be non empty sets. The cross product of A and B , denoted by A x B is the set of all ordered pairs ( x , y ) such that x A and y B . In symbols, A x B = {( x , y ) x A and y B}. The ordered pairs are also called points of the cross product. For example, if A = {1, 2, 3} and B = {a, b} then A x B = {(1, a), (1, b), (2, a), (2, b), (3, a), (3, b)}. The graph of A x B is given below: Note that if A and B are both equal to R , the set of real numbers, then A x B = R 2 = {( x , y ) x R and y R}. This is exactly the Cartesian pla ne or the coordinate plane . The plane can be obtained by drawing two perpendicular real number lines that intersect at the origin O. The horizontal line is the x-axis and the vertical
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line, the y-axis . In a coordinate plane, each point represents uniquely an ordered pair of real numbers (x, y) and each ordered pair of real numbers is represented by a point. If the point P = (x, y) then P is x units away from the y-axis and x units away from the x-axis. We say x is the abscissa and y is the ordinate of P. Taken together, x and y are called coordinates of P. Thus, if P = (3, 2) then P is 3 units to the right of the y-axis and 2 units above the x- axis. Our convention is that if x > 0, the point is x units to the right of the y-axis and if x < 0, the point is (-x) units to the left of the y-axis. Similarly, if y > 0, the point is y units above the x-axis and if y < 0, the point is (-y) units below the x-axis. The axes divide the plane into four quadrants as can be seen in the figure below. TIME TO THINK 1) Define each quadrant and axis in terms of the characteristics of the points that compose it. 2) Define distance between two points on a vertical line; on a horizontal line; in the plane. 3) Draw the straight line joining P 1 (2, -3) and P 2 (3, -2). 4) Define slope of a line. Find the slope of the line in #3. Describe the line whose slope is 0; positive; negative.
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