3 Functions
3.1. The Notion of a Function We already have a general notion of a function, since we have drawn straight line, quadratic and other curves.
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Figure 3.1 A graph is a diagram which represents a rule. To every value a, the rule assigns a new number, b, say. As a varies, so does the value of b. The graph is the collection of all points (a, b), where each b has been determined from a by the conversion rule. To indicate that b depends on a, we often write b = f (a) to indicate that a conversion rule, which we are calling f , has acted on a to obtain b. The fact that this collection of points often forms a smooth curve is a reflection of the type of conversion rule; it is not always the case, since it possible to write down some pretty wild conversion rules! If we know the specific form of the conversion rule, and have an explicit formula which tells us what f (a) actually is, then the graph is unnecessary, since all the information contained in the graph is described by that formula. Thus, rather than consider the quadratic graph y = x2 −3x+1, we could consider the function f (x) = x2 − 3x + 1. For Interest Instead of writing f (x) = x2 − 3x + 1, some authors write f : x 7→ x2 − 3x + 1, which can be read as “the function f such that x maps to x2 − 3x + 1. There is no difference in meaning between these two notations.
A function can be thought of as a “black box" — if you put in an input value x, out comes a definite output value. If we decide to label this particular function/“black box" by calling it f , then the output value is f (x). The output value f (x) is called the image of x under f . 34