Formalizing Relations and Functions
F.IF.1 Understand that a function from one set (called the domain) to another set (called the range) assigns each element of the domain exactly one element of the range... Also F.IF.2, F.IF.5
Goals for this section:
Students will learn what a domain and a range is and how to determine if a relation is a function or not.
Relation:
A relation is the pairing of two numbers with domain and range.
- Domain: all the x-values
- Range: all the y-values
When is a relation a function?
For every input (x-value), there should be only one output (y-value)!
- Example: If we plugged in the value x = 1 and the relation gave me back the answers y = 2 and y = 3, then it is not a function. This input had two outputs!
Method 1: Examining ordered pairs
Example: We have the ordered pairs: {(0,1), (4,3), (5,7), (6,2)}.
Step 1: List your domain and range.
Step 2: Are there any repeating x-values? If there are repeating x-values, it is not a function.
Example: We have the ordered pairs {(5,2), (4,1), (3,2), (7,3), (8,6)}
Step 1: List your domain and range.
Step 2: Are there any repeating x-values?
Example: We have the ordered pairs {(4,3), (9,2), (5,6), (3,1), (4,5), (8,4)}
Step 1: List your domain and range.
Step 2: Are there any repeating x-values?
Method 2: Vertical Line test
This is another method to determine whether a relation is a function. We take a straight line going up and down, pass it through the function, and if that line hits more than one point at a time then it is not a function.
- Use a pencil or a straight edge and move it left to right on the graph. Does the graph hit the pencil/straight edge more than once? Then it is not a function.
Example
The relation is represented by the black graph.
When doing the vertical line test, we can see how each vertical line passing through the original black line only passes through one point. This means that the relation is a function.
Function Notation:
f(x): Read as "f of x"
- It's just a name!
- Replace y with f(x)
We have seen functions with the variables x and y. However, there is another way we can write a function represented as f(x).
- Example: y=-3x+1 in function notation will be f(x)=-3x+1. We replace y with f(x) which depends on the x variable.
- Not only can we replace y with f(x), but we can also replace it with other similar notations such as g(x) and h(x).
Examples using function notation.
Find f(3) + g(2) given
- f(x)=3x-1
- g(x)= x2 +2.
Notice we don't have an x inside the parentheses. We have f(3) and g(2). This means to replace all instances of x with the number inside the parentheses for that function.
- For f(x), replace all x's with 3 and solve.
- For g(x), replace all x's with 2 and solve.
- Then add the two functions together since we're asked to do f(3) + g(2).
Answer:
External Resources
An explanation for understanding the difference between a relation and a function.