# Representing a Function Verbally and with Tables

🏆Practice representations of functions

Function, describes a correlation or coincidence between a dependent variable ($Y$) and an independent variable ($X$). The legitimacy of this relationship between the variables is called the " correspondence rule ".

## Verbal representation of a function

The verbal representation of a function expresses the connection between variables verbally, i.e. through a story.

A typical verbal representation of a function can look like this:

• Assuming that Daniel reads all the books he buys that month, the total number of books Daniel reads per year ($Y$) is a function of the number of books Danny buys each month ($X$).

#### Tabular representation of a function

A tabular representation of a function is a demonstration of the legitimacy of a function using a table of values $X$ (independent variable) and the corresponding values $Y$ (dependent variable).

In general, a table of values is shown as follows:

## Test yourself on representations of functions!

Is the given graph a function?

Examples of exercises on verbal and tabular representation of a function

### Example 1

$Y$ is a function of $X$ that corresponds to any value of $X$a number less than it in $2$.

$Y=X-2$

Solve the equation for each of the numbers of $X$ represented in the following table and place the correct number in. $Y$.

If $X = 1$, then Y will be equal to ____________

If $X = 13$, then Y will equal ___________ .

A value of $X$ corresponding to $Y = 10,5$ is __________

A value $X$ corresponding to $Y = 0$ is __________

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## Practice of verbal and tabular representation of a function

### Example 2

Describe in words the relationship between $X$ e $Y$.

### Example 3

Write down which table represents a function and which table does not represent a function

Do you know what the answer is?

### Example 4

$Y$ is a function of $X$ that corresponds to any value of $X$a number that is $5$ times greater than it.

$Y=5X$

Complete the table of values

• If $X = 1$, then $Y$ will equal ____________
• If $X = 13$, then $Y$ will be equal to ____________
• If $Y = 10,5$, then $X$ will be equal to ____________
• If $Y = 0$, then $X$ will be equal to ____________

### Example 5

$Y$ is a function of $X$ that corresponds to any value of $X$ a number $4$ times less than it.

#### $Y=\frac{X}{4}$

Complete the table of values

• If $X = 1$, then $Y$ will be equal to ____________
• If $X = 13$, then $Y$ will be equal to ____________
• If $Y = 10,5$, then $X$ will be equal to ____________
• If $Y = 0$, then $X$ will be equal to ____________

### Example 6

Complete the following table

• If $X = 10$, then $Y$ will equal ____________
• If $X = -2$, then $Y$ will be equal to ____________
• If $Y = 100$, then $X$ will be equal to ____________
• If $Y = -20$, then $X$ will be equal to ____________

### Example 7

Answer the following questions (for each example, write a table of values and draw the graph)

• Write an example of a function whose graph describes it as a continuous graph.
• Write an example of a function whose graph is discrete.

Do you think you will be able to solve it?

### Example 8

The function $Y$ corresponds to any number $X$ that is its root.

#### $Y=\sqrt{X}$

Complete the table of values

• If $X = 1$, then $Y$ will be equal to ____________
• If $X = 13$, then $Y$ will be equal to ____________
• If $Y = 10,5$, then $X$ will be equal to ____________
• If $Y = 0$, then $X$ will be equal to ____________

### Example 9

The function corresponds to any number $X$ less than $3$ of half the number

#### $Y=\frac{X}{2}-3$

• Complete the table of values
• If $X = 1$, then $Y$ will be equal to ____________
• If $X = 13$, then $Y$ will be equal to ____________
• If $Y = 10,5$, then $X$ will be equal to ____________
• If $Y = 0$, then $X$ will be equal to ____________

### Example 10

The function $Y$ corresponds to any number $X$ greater in $5$ times the number

#### $Y=2X+5$

• Complete the table of values

• If $X = 1$, then $Y$ will be equal to ____________
• If $X = 13$, then $Y$ will be equal to ____________
• If $Y = 10,5$, then $X$ will be equal to ____________
• If $Y = 0$, then $X$ will be equal to ____________

## Review questions

What is a function?

A function is a relationship between two variables, the variable $Y$ which is called the dependent variable, and the variable $X$, which is called the independent variable, between these two variables there is a correspondence rule; that is, for each value of $X$ there is only one value of $Y$.

What are the ways of representing a relationship?

A function can be represented as follows:

• Verbally
• Algebraically
• Table of values (Tabular)
• Graphically.

How to represent functions step by step?

Let's see an example of how to represent a function.

Example:

Represent the following function in its different forms

• Verbally:

Let $Y$ be a function of $X$ such that a value of $X$ corresponds to a number increased by $4$

• Algebraically:

$Y=X+4$

• Table of values:

We already have the function in algebraic form, now we are going to give values to $X$, to find the value of $Y$, according to the correspondence rule, and these values we are going to register them in a table:

Now we are going to substitute the values of $X$, to register the value that corresponds to $Y$, let's start with

We already know that the algebraic expression of this function is:

$Y=X+4$

Then,

When $X=-4$

$Y=-4+4=0$

When $X=-3$

$Y=-3+4=1$

When $X=-1$

$Y=-1+4=3$

When $X=0$

$Y=0+4=4$

When $X=2$

$Y=2+4=6$

When $X=5$

$Y=5+4=9$

According to this data we are now going to record it in the table

We have represented the function in a table of values.

• Graphically:

Finally we are going to represent these values in a graph:

We are going to find the pairs of coordinates in the Cartesian plane, and join each point as follows.

We can see that the function in graphical form is a linear function because it forms a straight line.

Do you know what the answer is?

## Examples with solutions for Representing a Function Verbally and with Tables

### 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

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

We should note that for every X value found on the graph, there is one and only one corresponding Y value.

Therefore, the graph is indeed a function.

Yes

### Exercise #3

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

We should note that for every X value found in the graph, there is one and only one corresponding Y value.

Therefore, the graph is indeed a function.

Yes

### Exercise #4

Determine whether the following table represents a function

### Step-by-Step Solution

It is important to remember that a constant function describes a situation where as the X value increases, the function value (Y) remains constant.

In the table, we can see that there is a constant change in the X values, specifically an increase of 2, and the Y value remains constant.

Therefore, according to the rule, the table describes a constant function.

Yes

### Exercise #5

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:

$y=-x-2$

$y=-x-2$