132 CHAPTER 2 Functions and Their Graphs Things to Know Function (pp. 63–65) • A relation between two nonempty sets so that each element x in the first set, the domain, has corresponding to it exactly one element y in the second set. The range is the set of images of the elements in the domain. • A function can also be described as a set of ordered pairs x y , ( ) in which no first element is paired with two different second elements. Function notation (pp. 65–67) • y f x( ) = • f is a symbol for the function. • x is the argument, or independent variable. • y is the dependent variable. • f x( ) is the value of the function at x. • A function f may be defined implicitly by an equation involving x and y or explicitly by writing y f x . ( ) = Difference quotient of f (pp. 68–69) f x h f x h h 0 ( ) ( ) + − ≠ Domain (pp. 69–71) If unspecified, the domain of a function f defined by an equation is the largest set of real numbers for which f x( ) is a real number. Vertical-Line Test (p. 78) A set of points in the xy-plane is the graph of a function if and only if every vertical line intersects the graph in at most one point. Even function f (p. 87) f x f x ( ) ( ) − = for every x in the domain ( x− must also be in the domain). Odd function f (p. 87) f x f x ( ) ( ) − = − for every x in the domain ( x− must also be in the domain). Increasing function (p. 89) A function f is increasing on an interval I if, for any choice of x1 and x2 in I, with x x , 1 2 < then f x f x . 1 2 ( ) ( ) < Decreasing function (p. 89) A function f is decreasing on an interval I if, for any choice of x1 and x2 in I, with x x , 1 2 < then f x f x . 1 2 ( ) ( ) > Constant function (p. 89) A function f is constant on an interval I if, for all choices of x in I, the values of f x( ) are equal. Local maximum (p. 91) A function f, defined on some interval I, has a local maximum at c if there is an open interval in I containing c so that f c f x , ( ) ( ) ≥ for all x in this open interval. The local maximum value is f c . ( ) Cube function (p. 103) f x x3 ( ) = x y 4 4 24 (1, 1) (0, 0) (21, 21) 24 Square root function (p. 103) f x x ( ) = x y 5 2 21 (1, 1) (0, 0) (4, 2) Cube root function (p. 103) f x x 3 ( ) = (1, 1) (21, 21) (28, 22) (8, 2) (0, 0) ( , ) 1 – 8 1 – 2 ( 2 ,2 ) 1 – 8 1 – 2 x y 3 28 23 8 Reciprocal function (p. 103) f x x 1 ( ) = x y 2 2 (1, 1) (21, 21) 22 22 Absolute value function (p. 104) f x x ( ) = x y 3 3 23 (1, 1) (0, 0) (21, 1) (2, 2) (22, 2) Greatest integer function (p. 104) f x x int ( ) ( ) = x y 4 2 22 2 4 23
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