# Ways to Represent a Function

🏆Practice representations of functions

## Ways to represent a function

### Algebraic Representation

Representation using an equation of $X$ and $Y$

### Graphical representation

Representation using a graph, plotting on the $X$ and $Y$ axis

### Tabular representation

Representation using a table $X,Y$ of points on the graph

### Verbal representation

Expressing the relationship between $X$ and $Y$ using words

### Function notation

$Y=$ or $f(x)=$

## Test yourself on representations of functions!

Is the given graph a function?

## Ways to represent a function

### Algebraic representation of 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.

### Graphical 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:

 $X$ $Y$ 0 -2 $1$ $-1$ 2 $4$

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.

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### Tabular representation of a function

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:

 $Y$ $X$ $-1$ $​ 0​$ $3$ $1$ $7$ $2$ $11$ $3$ $15$ $4$

### Verbal representation of a function

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 ($Y$)

$Y=3X$

Do you know what the answer is?

### Function notation

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 $X$.

## Examples with solutions for Representations of Functions

### Exercise #1

Is the given graph a function?

### Step-by-Step Solution

It is important to remember that a function is an equation that assigns to each element in domain X one and only one element in range Y

Let's note that in the graph:

$f(0)=2,f(0)=-2$

In other words, there are two values for the same number.

Therefore, the graph is not a function.

No

### Exercise #2

Which of the following equations corresponds to the function represented in the graph?

### Step-by-Step Solution

Use the formula for finding slope:

$m=\frac{y_2-y_1}{x_2-x_1}$

We take the points:

$(0,-2),(-2,0)$

$m=\frac{-2-0}{0-(-2)}=$

$\frac{-2}{0+2}=$

$\frac{-2}{2}=-1$

We substitute the point and slope into the line equation:

$y=mx+b$

$0=-1\times(-2)+b$

$0=2+b$

We combine like terms:

$0+(-2)=b$

$-2=b$

Therefore, the equation will be:

$y=-x-2$

$y=-x-2$

### Exercise #3

Which of the following equations corresponds to the function represented in the table?

### Step-by-Step Solution

We will use the formula for finding slope:

$m=\frac{y_2-y_1}{x_2-x_1}$

Let's take the points:

$(-1,4),(3,8)$

$m=\frac{8-4}{3-(-1)}=$

$\frac{8-4}{3+1}=$

$\frac{4}{4}=1$

We'll substitute the point and slope into the line equation:

$y=mx+b$

$8=1\times3+b$

$8=3+b$

Let's combine like terms:

$8-3=b$

$5=b$

Therefore, the equation will be:

$y=x+5$

$y=x+5$

### Exercise #4

Is the given graph a function?

No

### Exercise #5

Is the given graph a function?