Functions: The domain and range

Transcription

1 Mathematics Learning Centre Functions: The domain and range Jackie Nicholas Jacquie Hargreaves Janet Hunter c 6 Universit of Sdne

2 Mathematics Learning Centre, Universit of Sdne Functions In these notes we will cover various aspects of functions. We will look at the definition of a function, the domain and range of a function, and what we mean b specifing the domain of a function.. What is a function?.. Definition of a function A function f from a set of elements X to a set of elements Y is a rule that assigns to each element in X eactl one element in Y. One wa to demonstrate the meaning of this definition is b using arrow diagrams. X f Y 5 X g Y 5 6 f : X Y is a function. Ever element in X has associated with it eactl one element of Y. g : X Y is not a function. The element in set X is assigned two elements, 5 and 6 in set Y. A function can also be described as a set of ordered pairs (, ) such that for an -value in the set, there is onl one -value. This means that there cannot be an repeated -values with different -values. The eamples above can be described b the following sets of ordered pairs. F = {(,5),(,),(,),(,)} is a function. G={(,5),(,),(,),(,),(,6)} is not a function. The definition we have given is a general one. While in the eamples we have used numbers as elements of X and Y, there is no reason wh this must be so. However, in these notes we will onl consider functions where X and Y are subsets of the real numbers. In this setting, we often describe a function using the rule, = f(), and create a graph of that function b plotting the ordered pairs (, f()) on the Cartesian Plane. This graphical representation allows us to use a test to decide whether or not we have the graph of a function: The Vertical Line Test.

3 Mathematics Learning Centre, Universit of Sdne.. The Vertical Line Test The Vertical Line Test states that if it is not possible to draw a vertical line through a graph so that it cuts the graph in more than one point, then the graph is a function. This is the graph of a function. All possible vertical lines will cut this graph onl once. This is not the graph of a function. The vertical line we have drawn cuts the graph twice... Domain of a function For a function f : X Y the domain of f is the set X. This also corresponds to the set of -values when we describe a function as a set of ordered pairs (, ). If onl the rule = f() is given, then the domain is taken to be the set of all real for which the function is defined. For eample, = has domain; all real. This is sometimes referred to as the natural domain of the function... Range of a function For a function f : X Y the range of f is the set of -values such that = f() for some in X. This corresponds to the set of -values when we describe a function as a set of ordered pairs (, ). The function = has range; all real. Eample a. State the domain and range of = +. b. Sketch, showing significant features, the graph of = +.

4 Mathematics Learning Centre, Universit of Sdne Solution a. The domain of = + is all real. We know that square root functions are onl defined for positive numbers so we require that +, ie. We also know that the square root functions are alwas positive so the range of = +is all real. b. The graph of = +. Eample a. A parabola, which has verte (, ), is sketched below. 6 8 b. Find the domain and range of this function. Solution The domain of this parabola is all real. The range is all real. Eample Sketch the graph of f() = and find a. the domain and range b. f(q) c. f( ).

5 Mathematics Learning Centre, Universit of Sdne Solution The graph of f() =. a. The domain is all real. The range is all real where.5. b. f(q) =q q c. f( )=( ) ( ) = Eample The graph of the function f() =( ) + is sketched below. 6 The graph of f() =( ) +.

6 Mathematics Learning Centre, Universit of Sdne 5 State its domain and range. Solution The function is defined for all real. The verte of the function is at (, ) and therfore the range of the function is all real.. Specifing or restricting the domain of a function We sometimes give the rule = f() along with the domain of definition. This domain ma not necessaril be the natural domain. For eample, if we have the function = for then the domain is given as. The natural domain has been restricted to the subinterval. Consequentl, the range of this function is all real where. We can best illustrate this b sketching the graph. The graph of = for.. Eercises. a. State the domain and range of f() = 9. b. Sketch the graph of = 9.. Sketch the following functions stating the domain and range of each: a. = b. =

7 Mathematics Learning Centre, Universit of Sdne 6 c. = d. =.. Eplain the meanings of function, domain and range. Discuss whether or not = is a function.. Sketch the following relations, showing all intercepts and features. State which ones are functions giving their domain and range. a. = b. = c. = d. =, e. =. 5. Write down the values of which are not in the domain of the following functions: a. f() = b. g() =. Solutions to eercises.. a. The domain of f() = 9 is all real where. The range is all real such that. b. The graph of f() = 9.

8 Mathematics Learning Centre, Universit of Sdne 7. a. b. 5 The graph of =. The domain is all real and the range is all real. The graph of =. Its domain is all real and range all real. c. 6 8 The graph of =. The domain is all real and the range is all real. d. The graph of =. The domain is all real, and the range is all real.. = is not a function. If =, then = and =or =.

9 Mathematics Learning Centre, Universit of Sdne 8. a. b. The graph of =. This is a function with the domain: all real such that and range: all real such that. The graph of =. This is not the graph of a function. c. d. e. The graph of =. This is a function with the domain: all real and range: all real. The graph of =. This is the graph of a function which is not defined at =. Its domain is all real, and range is = ±. The graph of =. This is not the graph of a function. 5. a. The values of in the interval << are not in the domain of the function. b. = and = are not in the domain of the function.