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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.