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Lesson: Functions and Their Graphs
Objectives: 1. Determine whether a relation represents a function;
- Evaluate a function for a given input;
- Find the domain of a function
- Operate with functions
- Interpret the graph of a function
Part 1: Relations and Functions
A relation is a correspondence between two sets.
If x and y are two elements in these sets and if a relation exists between x and y , then we say that x corresponds to y or that y depends on x , and we write x y.
A relation may be represented as a diagram, using set-notation or by an equation/expression.
Ex 1: Diagram
Set notation: {(Arizona,9), (California,53), (Florida,27), (New York,27),(South Carolina,7), (Texas,36)}.
Ex 2: Diagram
Set notation: {(Dog,11), (Duck,10), (Rabbit,7)}.
State: Arizona California Florida New York South Carolina Texas
No of representatives 7 9 27 36 53
Animal:
Dog
Duck
Rabbit
Life Expectancy:
7
10
11
If X and Y are two non-empty sets, then a function from X to Y is a relation that associates with each element of X exactly one element of Y.
The set X is called the domain of the function.
The elements of the set X are called inputs, and the corresponding elements in Y are called outputs. The domain is also the set of all the inputs, and the set of all the outputs is called the range of the function.
The standard notation for a function is: ݂ ݂, ܻ→ ܺ: ሺݔሻ ൌ ݕ.
The Examples 1 and 2 above are functions.
Ex 3: Determine whether each relation represents a function. If it is a function, state the domain and range: a. {(2, 3), (4, 1), (3, -2), (2, -1)} Not a function; b. {(-2, 3), (4, 1), (3, -2), (2, -1)} Yes. Domain = {-2,2,3,4}, Range = {-2,-1,1,3}; c. {(2, 2), (4, 3), (3, 3), (8, -1)} Yes. Domain = {2,3,4,8}, Range = {-1,2,3}.
Ex 4: Determine if the equation ݕൌ 2 ݔ 3 defines y as a function of x. For each value we choose for the input x, there is only one value for y. The answer is yes.
Ex 5: Determine if the equation ݕ2 ൌ ݔ ଶ^ 3 defines y as a function of x.
The expression of y is: ݕൌ േට ௫ିଷ ଶ. For the value x = 5, we get two different values for y:^ േ. The answer is no.
Part 4: Operations with functions
Sum of f + g
(f + g)(x) = f(x) + g(x)
Ex 11 : ݂ ሺݔሻ ൌ ଶ௫ିଵ ௫ ,^ ݃
a). the domain of the function f: R {0}. b). the domain of function g: R {-1}. c). ݃ ݂ሺ ሻሺݔሻ ൌ݂ ݃ ሻݔሺ ሺݔሻ ൌ ଶ௫ିଵ ௫ ^
௫ ௫ାଵ ൌ^
ሺଶ௫ିଵ ሻሺ௫ାଵሻା௫௫ ௫ሺ௫ାଵሻ ൌ^
ଷ௫ మ^ ା௫ିଵ ௫ మ^ ା௫. d). ݃െ ݂ሺ ሻሺݔሻ ൌ݂ ݃െ ሻݔሺ ሺݔሻ ൌ ଶ௫ିଵ ௫ െ^
௫ ௫ାଵ ൌ^
ሺଶ௫ିଵ ሻሺ௫ାଵሻି௫௫ ௫ሺ௫ାଵሻ ൌ^
௫ మ^ ା௫ିଵ ௫ మ^ ା௫. e). ݃∙ ݂ሺ ሻሺݔሻ ൌ݂ ݃∙ ሻݔሺ ሺݔሻ ൌ ଶ௫ିଵ ௫
௫ ௫ାଵ
ଶ௫ିଵ ௫ାଵ
f). ቀ
ሺ௫ሻ ሺ௫ሻ
మೣషభೣೣೣ
శభ
ଶ௫ିଵ ௫
௫ାଵ ௫
ଶ௫ మ^ ା௫ିଵ ௫ మ^
Part 5: The Graph of a function
Vertical Line Test : A set of points in a xy -plane is the graph of a function if and only if every vertical line intersects the graph in at most one point.
Ex 12: Functions
Difference of f - g
(f - g)(x) = f(x) - g(x)
Product of f g
(f g)(x) = f(x)g(x)
Quotient of f/ g
(f /g)(x) = f(x)/g(x)
Ex 13: Not functions
Ex 14:
a. What are ݂ ሺ0ሻ,݂ ቀ ଷగ ଶ ݂, ቁ^ ሺ3ߨሻ?
From the graph, ݂ ሺ0ሻ ൌ 4,݂ ቀ ଷగ ଶ ቁ ൌ 0,݂^
b. The domain of f is ሾ0,4ߨሿ. c. The range of f is [-4,4]. d. The y-intercept is (0,4). The x-intercepts are: ቀ గ ଶ , 0ቁ , ቀ^
ଷగ ଶ , 0ቁ , ቀ^
ହగ ଶ , 0ቁ , ቀ^
గ ଶ , 0ቁ. e. The function is positive (f(x)>0) for ݔ∈ ቀ0, గ ଶ ቁ ∪ ቀ^
ଷగ ଶ ,^
ହగ ଶ ቁ ∪ ቀ^
గ ଶ , 4ߨቁ.
Ex 15: ݂ ሺݔሻ ൌ ௫ ௫ିଵ. a. Domain of the function: because the equation ݔെ 1 ൌ 0 has the solution x=1, then the domain will be ሼ ݔ⋮ ݔ് 1ሽ , or R {1}. b. Is the point (2,2) on the graph? If x=2 is replaced, then we get ݂ ሺ2ሻ ൌ ଶ ଶିଵ ൌ 2 , so that the point (2,2) is on the graph. c. If x=4, then ݂ ሺ4ሻ ൌ ସ ସିଵ ൌ^
ସ ଷ,^ so that the point^ ቀ4,^
ସ ଷቁ^ is on the graph of f. d. If f(x)=5, then ௫ ௫ିଵ ൌ 5 → ݔ ൌ^
ହ ସ , so the point^ ቀ
ହ ସ , 5ቁ^ is on the graph of f.
