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Graphs of functions, and algebra FUNCTIONS: As you remember, a function is a rule that takes in one number and gives out another. It’s a special case of a relation: a relation is a function if every element in the domain relates to precisely one element of the codomain, so it makes sense to talk about “the element f(x) related to x” A partial function has each element in the domain related to at most one element in the domain. We can talk about “the element, if there is one , related to x” NOTATION: a reminder If we introduce a dummy variable in referring to a function, it needs to do something. Rigorous: "the sine function" "the function mapping x to sin(x)" “the function mapping x to x3+3x-1” Casual: "The function y = sin(x)" (depends on a convention that "x" is usually the domain and “y” the codomain; this can be tricky when talking about inverses, composites, and implicit functions (see below)) Sloppy: "the function sin(x)" (the role of x is never made clear) MORE NOTATION: “The equation y = sin(x)” is fine. An equation may involve any number of variables and does not distinguish between “domain” and “codomain”. IF an equation implies a unique y value for every x, or a unique x value for every y, these are called “implicit functions”. x3 = y5 generates an implicit function x x3/5 and an implicit function y y5/3. x2 =y2 does not generate an implicit function (unless we restrict the domain and codomain). GRAPHS A relation from the real numbers to the real numbers can (if it's simple enough) be represented by a graph. Perpendicular (horizontal and vertical) axes are drawn. Every point in the plane is given real coordinates (x,y) representing horizontal and vertical distance. The plane is thus the product of two copies of the real numbers. Every subset of the plane represents a relation R R. This is exactly like a relation table but for infinitely many values. Draw and label a pair of axes. Plot the points (0,0), (0,1), (0,2) with a solid dot. Plot the points (1,0), (2,0), (-1,0) with a small ring. Plot the point (-1,1) with a star. A square has two opposite corners at (1,2) and (-1,-2). Draw the square and find the coordinates of the other two vertices. Draw and compare on (a) the reals and (b) the set {0,1,2,3,4}: (1) the relation x=y (2) the relation x>y (3) the function or partial function taking x to x+1 (4) the function or partial function taking x to x2 THE VERTICAL LINE TEST: A set of points in the plane is the graph of a function if every vertical line meets the set in exactly one point. A table represents a function if every column contains exactly one entry. A set of points in the plane is the graph of a partial function if every vertical line meets the set in at most one point. A table represents a partial function if Draw a set of axes and plot some points (x,y) for which y=1. Guess what the set of all such points is, and draw as much of it as you can. Does the y value for one of these sets depend on x? Does this graph describe a relation ? Does this graph describe a function ? On the same axes, graph y=2 and y=-1. Draw a set of axes and plot some points (x,y) for which y = x+1. Then sketch in the set of all such points. Do the same thing for y = x and y=x-2. What do they have in common? Draw a set of axes and plot some points (x,y) for which y = x+1. Then sketch in the set of all such points. Do the same thing for y = 2x+1 and y= 1-x. What do they have in common? Is it true that every straight line is the graph of a function? Which straight lines are the graphs of equations of the form y = mx + b ? What do these lines have in common? The value m represents the slope. Positive slopes rise from left to right, negative slopes sink from left to right . I The value b represents the y-intercept – the point at which x=0. If the x-intercept is the point (if any) at which y=0, what are its coordinates? Which lines don’t have x-intercepts? Plot the relations: x+y=1 x + 2y = 1 x - 2y = 2 3x + 2y = 6 Are these functions? Can you guess the slope and intercepts of 2x + 3y = 12 ; x–y=2 ? Check your answers by plotting. The slope of ax + by = c is: The x-intercept is: The y-intercept is: