What Is A Function?

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Functions
Many equations describe a real-life relationship between two quantities, also known as function. The function is said to be the central idea in the study of mathematics.

A \(\underline{function}\) is a specific rule that compares one quantity to another quantity. The correspondence between two variables, the x and y, use a set of ordered pairs (x, y) to show that the x is related to y. The interrelation between number patterns or ordered pairs can be represented, graphs, and equations. An example of a function is one that shows the relationship between the employee salary (x) to the number of hours work (y). Another example is (x) water bill paid for a month and its relationship to the cubic meters of water consumed (y) plus other charges (k).
Example:
Observe the following ordered pair
a. {(1, 2), (2, 3), (3, 4), (4, 5)}
b. {(2, 3), (1, 5), (3, 7), (3, )}

What have you noticed?

a. {(1, 2), (2, 3), (3, 4), (4, 5)}
We can compare it this way
x            y
1            2
2           3
3           4
4           5

The relation between the ordered pair in this example is a function. Looking at the value of x, there is no ordered pair with the same first element.

b. {(2, 3), (1, 5), (3, 7), (3, 9)}
x           y
2          3
1          5
3          7
3         9

This is not a function since the first element, which is 3, corresponds to the different second elements, which are 7 and 9.

Function Notation
Function notation is the functional relationship between two variables such x and y. It is represented by this equation:
y = f(x)       Read as y equals function of x, or y is a function of x. It means that the value of y depends on the value of x. Thus, x is an independent variable while y is a dependent variable.

Example:
Given the f(x) = 2x or y = 2x, find f (3) and f(2).
y = 2x                                      The function means that x is multiplied by 2 to get the value of y
f(3)
y = 2x
y = 2(3)                                   Substitute the value of function
y = 6

f(2)y = 2x
y = 2(2)
y = 4

The functions of x to y are (3, 6) and (2, 4).

Graph of Functions
Graphs are used to visualize the relationship between two quantities. Function relationships between the function of x to y can be represented by a graph.

Example:
\( f(x)  = x^2\)
To sketch a graph, determine the ordered pairs. The easy way to do this is to set up a table and plot the points.

Solve for all the value of y.
\( y =  {(-2)}^2 = 4\)
\( y =  {(-1)}^2 = 1 \)
\( y =  {(0)}^2 = 0 \)
\(y =  {(1)}^2 = 1 \)
\( y  =  {(2)}^2 = 4 \)

We can now plot the ordered pair
The graph of the function \( f(x) =x^2 \) is a parabola with a vertex of (0,0) and the axis of symmetry is the y-axis.