2-1: Relations and Functions :
2-1: Relations and Functions **If ever you need a hint click on me!!!! Think of me as your own personal tour guide. 2-1: Relations and Functions :
2-1: Relations and Functions Objectives:
1) To graph relations
2) To identify domain and range given ordered pairs, a table, and a graph
3) To identify functions using a mapping diagram
4) To identify functions given ordered pairs and by using vertical line test
5) To evaluate functions
6) To show understanding of above via quiz
7) To create or find and restate a real- world example of domain and range via journal reflection Objective 1: Graphing relations (Plot all points) :
Objective 1: Graphing relations (Plot all points) Graph the relation:{(2, -3), (1, 2), (2, 4), (0,0), (1, -1), (3, 0)} Relations & Functions :
Relations & Functions Relation: a set of ordered pairs
Domain: the set of x-coordinates
Range: the set of y-coordinates
When writing the domain and range, do not repeat values. Objective 2: Stating Domain and Range given ordered pairs :
Objective 2: Stating Domain and Range given ordered pairs Given the relation:
{(2, -6), (1, 4), (2, 4), (0,0), (1, -6), (3, 0)}
State the domain:
D: {0,1, 2, 3}
State the range:
R: {-6, 0, 4} Relations and Functions :
Relations and Functions Relations can be written in several ways: ordered pairs, table, graph, or mapping.
We have already seen relations represented as ordered pairs. Objective 2: Identify the domain and range Table :
Objective 2: Identify the domain and range Table {(3, 4), (7, 2), (0, -1),
(-2, 2), (-5, 0), (3, 3)} Objective 3: Identifying functions using a mapping diagram :
Objective 3: Identifying functions using a mapping diagram Create two ovals with the domain on the left and the range on the right.
Elements are not repeated.
Connect elements of the domain with the corresponding elements in the range by drawing an arrow. Mapping :
Mapping {(2, -6), (1, 4), (2, 4), (0, 0), (1, -6), (3, 0)} Objective 4: Identifying if a relation is a function given ordered pairs :
Objective 4: Identifying if a relation is a function given ordered pairs A function is a relation in which the members of the domain (x-values) DO NOT repeat.
So, for every x-value there is only one y-value that corresponds to it.
y-values can be repeated. Do the ordered pairs represent a function? :
Do the ordered pairs represent a function? {(3, 4), (7, 2), (0, -1), (-2, 2), (-5, 0), (3, 3)}
No, 3 is repeated in the domain.
{(4, 1), (5, 2), (8, 2), (9, 8)}
Yes, no x-coordinate is repeated. Objective 4: Identifying Functions using the Vertical Line Test :
Objective 4: Identifying Functions using the Vertical Line Test Vertical Line Test:
If a vertical line is passed over the graph and it intersects the graph in exactly one point, the graph represents a function. Objective 2 and 4: Does the graph represent a function? Name the domain and range. :
Objective 2 and 4: Does the graph represent a function? Name the domain and range. Yes
D: all reals
R: all reals
Yes
D: all reals
R: y ≥ -6 Objective 2 and 4: Does the graph represent a function? Name the domain and range. :
Objective 2 and 4: Does the graph represent a function? Name the domain and range. No
D: x ≥ 1/2
R: all reals
No
D: all reals
R: all reals Objective 2 and 4: Does the graph represent a function? Name the domain and range. :
Objective 2 and 4: Does the graph represent a function? Name the domain and range. Yes
D: all reals
R: y ≥ -6
No
D: x = 2
R: all reals Objective 5: Evaluating Functions using Function Notation :
Objective 5: Evaluating Functions using Function Notation When we know that a relation is a function, the “y” in the equation can be replaced with f(x).
f(x) is simply a notation to designate a function. It is pronounced ‘f’ of ‘x’.
The ‘f’ names the function, the ‘x’ tells the variable that is being used. Objective 5: Value of a Function :
Objective 5: Value of a Function Since the equation y = x - 2 represents a function, we can also write it as f(x) = x - 2.
Find f(4):
f(4) = 4 - 2
f(4) = 2 Objective 5: Value of a Function :
Objective 5: Value of a Function If g(s) = 2s + 3, find g(-2).
g(-2) = 2(-2) + 3
=-4 + 3
= -1
g(-2) = -1 Objective 5: Value of a Function :
Objective 5: Value of a Function If h(x) = x2 - x + 7, find h(2c).
h(2c) = (2c)2 – (2c) + 7
= 4c2 - 2c + 7 Objective 5: Value of a Function :
Objective 5: Value of a Function If f(k) = k2 - 3, find f(a - 1)
f(a - 1)=(a - 1)2 - 3
(Remember FOIL?!)
=(a-1)(a-1) - 3
= a2 - a - a + 1 - 3
= a2 - 2a - 2 Optional Examples (if you need more help)- Fading into the Homework-Fill in the blanks! :
Optional Examples (if you need more help)- Fading into the Homework-Fill in the blanks! 1. The domain= {-3, _, 1, _, 5} and the
range = {-2, _, _, 2, 4}
2. Finish the mapping diagram of this relation. Given the Relation {(1, -2), (0, 2), (-3, 4), (5,0), (1, -1), (3, 0)} Optional Examples Continued :
Optional Examples Continued 3. Is the relation a function. Why or why not? (Hint: A relation is NOT a function if the x values repeat).
Answer: No! The x value of 1 repeats in the domain.
4. Find f(3) for f(x) = -5x - 7.
First Step:
f(3)= - 5 (3) – 7 =
Second Step:
-5(3) – 7 = -15 – 7
Third Step with Answer:
-15 – 7 = -22
5. Find f(-1) for f(x) = x2 - x + 7.
First Step:
f(-1)= (-1)2 – (-1)+ 7.
Second Step:
(-1)2 – (-1)+ 7 = 1 + 1 + 7
Third Step with Answer:
1 + 1 + 7 = 9 Homework- Include in notes under examples :
Homework- Include in notes under examples 1.
2. State the domain and range of the previous relation.
3. Create a mapping diagram of this relation.
4. Is this relation a function. Why or why not?
5. Find f(3) for f(x) = 2x + 5.
6. Find f(-1) for f(x) = x2 - x + 7. Graph the relation:{(1, -2), (0, 2), (-3, 4), (5,0), (1, -1), (3, 0)} After completing all homework problems go back to module to complete last two Objectives. :
After completing all homework problems go back to module to complete last two Objectives. 6) To show understanding of above via quiz
7) To create or find and restate a real- world example of domain and range via journal reflection