Ways to Represent a Function

Before we talk about algebraic representation, it is important to understand what a function means.
A function describes the relationship between $X$ and $Y$.
In any function, $X$ is the independent variable and $Y$ is the dependent variable. This means that every time we change $X$, we get a different $Y$.
Y depends on $X$ and $X$ depends on nothing.

Important point: For each $X$ there will be only one $Y$!

An algebraic representation of a function is essentially the equation of the function.

Let's look at some examples of algebraic representation of a function and analyze them:
$Y=X-3$

In this equation, it is clear that $Y$ depends on the $X$ we substitute into the equation.
If $X=1$, then $Y=-2$
If $X=0$, then $Y=-3$
If $X=2$, then $Y=-1$
In other words, the relationship between $X$ and $Y$ is that $Y$ will always be $3$ less than $X$.

Now let's examine another equation:
$y=2x-5$

Also in this equation, it is clear that $Y$ depends on the $X$ we substitute into the equation.
If $X=3$, then $Y=1$
If $X=4$, then $Y=3$
If $X=5$, then $Y=-5$
In this equation, it is difficult to define in words the relationship between $X$ and $Y$, so we will say that the relationship between them is the equation itself:
$y=2x-5$

Now let's examine another equation:
$y=x$

In this equation, it is also clear that $Y$ depends on the $X$ we substitute into the equation.
If $X=3$, then $Y=3$
If $X=2$, then $Y=2$
If $X=1$, then $Y=1$
The relationship between $X$ and $Y$ is that they are identical each time.

Click here to learn more about the algebraic representation of a function!

A graphical representation of a function shows us how the function looks on the $X$ and $Y$ axes.
What is most important to know?
For each $X$, there is only one $Y$, and to draw a function as a graph, it is advisable to find at least 3 points of the function.
How to draw the function:
Each time, substitute a different $X$ into the algebraic representation and identify its $Y$. Mark all the points obtained on the drawing and then draw a straight line between them.

For example:
$Y=3X-2$
Let's substitute three $X$s and we get:

Now let's mark the points we obtained on the number line:

Examples of graphical representation of a function:

Important tips:
How do you know if the function is increasing or decreasing?
There are 2 ways:

1. According to the coefficient of $X$ in the algebraic representation – if the coefficient of $X$ is positive, the function increases from left to right. If negative, the function decreases from left to right.
2. Mark $3$ points of the function (substitute a different $X$ each time and find the $Y$) and then draw a straight line passing through them. Look from left to right and decide if the function is increasing or decreasing.

You can read more about the graphical representation of a function at this link!

A tabular representation is essentially a representation using a table of $X$ and $Y$ showing us the value of $Y$ for each $X$ that we substitute into the function.

Let's see an example:
For the algebraic representation - $Y=4X-1$
we get a tabular representation like this:

A verbal representation of a function describes the relationship between $X$ and $Y$ using words.

For example:
Each package of flour ($X$) makes $3$ whole pizzas ($Y$)

$Y=3X$

For more information on verbal and tabular representation of a function, click here!

How do we denote a function?
So far, we have denoted a function as $Y=……$
It is also useful to know that a function can be denoted in the following way:
$F(x)=……...$ which implies that we will get a value that depends on $X$.

You can read more about function notation here!