68 CHAPTER 2 Functions and Their Graphs Implicit Form of a Function In general, when a function f is defined by an equation in x and y, we say that the function f is given implicitly . If it is possible to solve the equation for y in terms of x, then we write ( ) = y f x and say that the function is given explicitly . For example, ( ) ( ) ( ) + = = = − + − = = = − = = = x y y f x x x y y f x x xy y f x x Implicit Form Explicit Form 3 5 3 5 6 6 4 4 2 2 4 Find the Difference Quotient of a Function An important concept in calculus involves using a certain quotient. For a function f, the inputs x and + ≠ x h h , 0, result in the values ( ) f x and ( ) + f x h . The quotient of their differences ( ) ( ) ( ) ( ) ( ) + − + − = + − f x h f x x h x f x h f x h with ≠ h 0, is called the difference quotient of f at x. SUMMARY Important Facts about Functions • For each x in the domain of a function f, there is exactly one image ( ) f x in the range; however, more than one x in the domain can have the same image in the range. • f is the symbol that we use to denote the function. It is symbolic of the equation (rule) that we use to get from x in the domain to ( ) f x in the range. • If ( ) = y f x , then x is the independent variable, or the argument of f, and y, or ( ) f x , is the dependent variable, or the value of f at x. DEFINITION Difference Quotient The difference quotient of a function f at x is given by ( ) ( ) + − ≠ f x h f x h h 0 (1) The difference quotient is used in calculus to define the derivative , which leads to applications such as the velocity of an object and optimization of resources. When finding a difference quotient, it is necessary to simplify expression (1) so that the h in the denominator can be divided out. Finding the Difference Quotient of a Function Find the difference quotient of each function. (a) ( ) = − f x x x 2 3 2 (b) ( ) = f x x 4 (c) ( ) = f x x EXAMPLE 8 COMMENT The TI-84 Plus CE graphing calculator requires the explicit form of a function. Desmos and Geogebra accept the implicit form. ■
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