Chapter 1
Functions and Limits
1.1
Functions
Definition 1 (Real-Vlaued Function of a Real Variable)() Let and be sets of real numbers. A Real-Valued function of a real variable (independent variable) from to is a correspondence that assigns to each number in exactly one number (dependent variable) in
() The domain of is the set . The number is the image of under and is denoted by () The range of is a subset of and consists of all images of numbers in
Example 1 () Equation in implicit form: 2+ 2 = 1; () Equation in explicit form
= 1 2 ¡ 1− 2¢; () Function notation () = 1 2 ¡ 1− 2¢
The original equation, 2 + 2 = 1 implicitly defines as a fuction of .
When you solve the equation for , you are writing the equation in explicit form.
The Domain and Range of a Function:
The domain of a function can be descrabed explicitly, or it may be de-scribed implicitly by an equation used to define the function. The implied domain is the set of all real number for which the equation is defined. Example 2 The domain of the function
() = 8 + 10 + 2
Example 3 Determine the domain and range of the function () =
½
1− if 1 √
1− if ≥ 1 The Graph of a Function:
The graph of the function = () consists of all points ( ()) where is in the domain of Note that
() = the directed distance from the −axis;
() () = the the directed distance from the −axis. Example 4 () = 8 + 10 + 2 15 12.5 10 7.5 5 2.5 0 15 12.5 10 7.5 5 2.5 0 0 0 0 0 Example 5 () = ½ 1− if 1 √ 1− if ≥ 1 2
5 2.5 0 -2.5 -5 5 2.5 0 -2.5 x y x y Constant Functions: ()≡constant Elementary functions:
Elementary functions fail into three categories: ()Algebraic function (polynomial, radical, rational).
()Trigonometric function (sine, cosine, tangent, and so on). ()Exponential and logarithmic.
Composite Functions:
You can combine two (elementary) functions in yet another way, called composition. The resulting function is called a composite function.
Definition 2 Let and be function. The function given by ( ◦ ) () = ( ()) is called the composite of with The domain of ◦ is the set of all in the domain of such that () is in the domain of
Example 6 Given () =√ and () = 5 − find ◦ Example 7 Let
() = 5 (2+ 9)3
Find two functions () and () whose composite is () () = 53 () =
2+ 9
Even and Odd functiuons:
Definition 3 (Even and Odd)() The function = () is even if () = (−)
() The function = () is odd if (−) = − () Example 8 () () = 3
− () () = 1 + cos 3
