Machinery's Handbook, 31st Edition
Functions
35
Functions Functions are understood through both equations and graphs. Graphs are drawn on the ( x , y )-coordinate system (or rectangular coordinate system), which is described fully in Analytic Geometry section of GEOMETRY starting on page 42. A function consists of two sets (generally called X and Y ) of numbers and a rule that assigns (or sends) each element (usually, a real number) x in X to a unique element (another real number) y in Y . “ y is a function of x ” is commonly expressed as y = f ( x ). Countless physical processes are represented by functions. For example, displacement, s , is a func- tion of time, t , so it is represented as s ( t ). Velocity also is a function of time, v ( t ). The set X is called the domain of a function, which is the set of all real number values for which a function is defined . When the function acts on these numbers, the result is another real number y . As an example, y = f ( x ) = 2 x + 5 is a linear function whose domain is the set of all real numbers, since when 5 is added to the product of 2 and any real number x , the result is always a real number y . Lines are first-degree polynomials; in fact, the domain of any polynomial is the set of reals (see Polynomials on page 28). Two more examples: f ( x ) = 1/ x is not defined at x = 0, since 1/0 is not a number, so its domain is the set of all real numbers except 0. Since negative numbers do not have square roots in the reals, the domain of f ( x ) = x is the set of non-negative real numbers. In y = f ( x ), x is called the input to the function. The y value resulting from x is the output , or function value . Since y depends on x , it is the dependent variable ; x is the independent variable . Interval Notation: Domain is often expressed in interval notation , indicating the por- tion of the number line that contains a function’s valid input values. The interval notation for set of all reals is ( , ) − ∞ ∞ ; for non-negative numbers, it is [0,∞); for positive numbers, it is (0,∞); and for all real numbers except 0, it is ( , ) ( , ). − ∞ 0 0 ∞ The domain of a poly- nomial is therefore ( , ) − ∞ ∞ ; the domain of f ( x ) = 1/ x is ( , ) ( , ) ; − ∞ 0 0 ∞ and the domain of f ( x ) = x is [0,∞). Graphs of Functions.—The graph of a function is drawn through the points ( x , y ) on the rectangular coordinate system (see page 42 ) so that the curve satisfies the equation y = f ( x ). A graph depicts the relationship between x and y . The graph of a function will always pass the vertical line test , by which any vertical line intersects the graph at most once. Thus, any line other than a vertical line is expressible in function form, f ( x ) = mx + b , where the slope is m and the y -intercept is b . (The several forms of linear equations are explained in Equation Forms of a Line on page 44.) A parabola is represented by a polynomial function of the form f ( x ) = ax 2 + bx + c . All polynomials pass the vertical line test (Fig. 1a).
Fig. 1. (a) Passes Vertical Line Test—Represents a Function; (b) Fails the Test—Does Not Represent a Function. Sketches of Basic Functions: Sketches of basic functions and domain interval notation are shown in Fig. 2. It is obvious that each function passes the vertical line test.
Copyright 2020, Industrial Press, Inc.
ebooks.industrialpress.com
Made with FlippingBook - Share PDF online