Function
The relation f with first set A and second set B is called a function when every element of A is paired with only one element of B. f:A→B
We can write it based on the way we label the sets. You may find it written as f:X→Y
If in the functional relation f with first set X and second set Y the element y1 ∈ X is paired with the element y1 ∈ Y then x1 we call it the face of y1 and y1 is the reflection of x1
In other words y1 is the range of the function f in x1 and we label it as f(x1).
For f:X→Y, the set of faces we call the domain of the function. The domain is the first set of the function.
The set of reflection (range) of the f:X→Y is a subset F of the second set Y.
The function f:X→Y where X and Y are subsets of the set of real numbers R are called numeric functions.
For a numeric function if the second set is not given it is understood that is the set of real numbers R. The numeric function can be demonstrated on a tabular form, with formula and diagram.
Example 1: In the figure below we have diagrams of three different relations with first set X = {1, -2, 3} and second set Y = {2, -4, 6, 7}.
a) Show which of those relations are functions.
Based on the function’s definition, the third diagram is a function because every element of the first set is paired with only one element of the second set.
b) For the functional relation, show the domain. Find for all faces the corresponding value of the function and show the range.
Domain: X = {1, -2, 3}
Range: Y = {2, -4, 6, 7}
c) Give it in a tabular form.
d) Give it with formula.
f:X→Y. Y = 2x
e) Illustrate the graph of this function on XOY
Example 2: The numeric function f with the tabular form:
a) Show the domain and the range.
Domain: X = {-3, -2, -1, 0, 1, 2}
Range: Y = {3, 2, 1, 0, 1, 2}
b) Find f(-3), f(0), f(2).
Based on the tabular form above we find.
f(-3) = 3
f(0) = 0
f(2) = 2
c) Illustrate the graph of this function on the OXY plan.
Note: This graph is the graph of the absolute value y=|x|
Don’t Forget:
When f(x) is an expression with a variable x, it is true for every value of x from the set E, E⊂R, pairing every value of x from E the corresponding value of f(x) [The value of the variable y we find it with the formula y=f(x)], we obtain a function with domain E. We label it:
f:y=f(x), x∈E or →f(x), x∈E
Example 3: The function f has the formula y = x³, x∈E. Where E = {-3, -2, -1, 0, 1, 2, 3}
a) Find f(-2), f(0), f(2).
f (-2) = (-2)³ = -8
f(0) = 0³ = 0
f (2) = 2³ = 8
b) Give this on a tabular form.
c) Illustrate its graph on the XOY
