Representing a Function Verbally and with Tables

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:

A1 - Verbal representation of a new function

Detailed explanation

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Test yourself on representations of functions!

Is the given graph a function?

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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. $Y$ .

verbal and tabular representation of a function

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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Test your knowledge

Question 1

Is the given graph a function?

Question 2

Determine whether the following table represents a function

Question 3

Determine whether the data in the following table represent a constant function

Practice of verbal and tabular representation of a function

Example 2

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

Describe in words the relationship between X and 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?

Question 1

Determine whether the following table represents a constant function

Question 2

Determine whether the following table represents a function

Question 3

Determine whether the following table represents a constant function:

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

a function of X corresponding to any value of X

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 ____________

Check your understanding

Question 1

Determine whether the given graph is a function?

Question 2

Does the graph below represent a function?

Question 3

Is the given graph a function?

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?

Question 1

Is the given graph a function?

Question 2

Is the given graph a function?

Question 3

Determine whether the following table represents a linear function

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 ____________

Test your knowledge

Question 1

Given the following graph, determine which table corresponds to the following table

Question 2

Given the following graph, determine which table corresponds to the following table

Question 3

Is the given graph a function?

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:

1 - How to represent functions step by step

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

2 - How to represent functions step by step

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.

3 - How to represent functions step by step

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

If you are interested in this article you may also be interested in the following articles:

Graphical representation of a function

Algebraic representation of a function

Notation of a function

Domain of a function

Indefinite integral

Numerical value assignment in a function

Increasing function

Decreasing function

Constant function

Growing and decreasing intervals of a function

Functions for seventh grade

In Tutorela you will find a variety of articles with interesting explanations about mathematics.

Do you know what the answer is?

Question 1

Is the given graph a function?

Question 2

Determine whether the following table represents a function

Question 3

Determine whether the data in the following table represent a constant function

Examples with solutions for Representing a Function Verbally and with Tables

Exercise #1

Determine whether the data in the following table represent a constant function

Video Solution

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 observe that there is a constant change in X values, meaning an increase of 1, and a non-constant change in Y values - sometimes increasing by 1 and sometimes by 4

Therefore, according to the rule, the table does not describe a function

Answer

Exercise #2

Determine whether the following table represents a function

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

Step 1: Identify the given pairs of $(X, Y)$ .
Step 2: Verify that each $X$ maps to exactly one $Y$ .
Step 3: Conclude whether the table represents a function.

Now, let's work through each step:
Step 1: The pairs given are: $(-2, 1)$ , $(2, 6)$ , $(6, 11)$ , $(10, 16)$ , $(14, 21)$ .

Step 2: For each input value $X$ , we check its corresponding output $Y$ :

$X = -2$ maps to $Y = 1$
$X = 2$ maps to $Y = 6$
$X = 6$ maps to $Y = 11$
$X = 10$ maps to $Y = 16$
$X = 14$ maps to $Y = 21$

None of the values of

X

is associated with more than one different

Y

value.

Step 3: Since each $X$ value has exactly one corresponding $Y$ value, the table represents a function.

Yes

Answer

Yes

Exercise #3

Is the given graph a function?

Video Solution

Step-by-Step Solution

To determine if the given graph represents a function, we use the vertical line test: if any vertical line intersects the graph at more than one point, the graph is not a function.

Let's apply this test to the graph:

Examine different sections of the graph by drawing imaginary vertical lines.
Look for intersections where more than one point exists on the vertical line.

Upon examining the graph, we observe that there are several vertical lines that intersect the graph at multiple points, particularly in areas with loops or overlapping curves. This indicates that at those $x$ -values, there are multiple $y$ -values corresponding to them.

Since there exist such vertical lines, according to the vertical line test, the graph does not represent a function.

Thus, the solution to this problem is that the given graph is not a function.

Answer

Exercise #4

Is the given graph a function?

Video Solution

Step-by-Step Solution

To determine whether the graph represents a function, we apply the Vertical Line Test. Here are the steps we follow:

Step 1: Visualize placing a vertical line across various parts of the graph.
Step 2: Check if the vertical line intersects the graph at more than one point at any given position.

Step 1: On evaluating the given graph carefully, there is a notable presence of a vertical line passing through multiple y-values. Specifically, the vertical line goes from $y = -3$ to $y = 3$ at $x = 3$ .

Step 2: Since this vertical line at $x = 3$ intersects the graph at an infinite number of points, it fails the Vertical Line Test.

Therefore, the graph does not represent a function. According to our analysis and the Vertical Line Test, the correct answer is No.

Answer

Exercise #5

Determine whether the following table represents a constant function

Video Solution

Step-by-Step Solution

To determine if the table represents a constant function, we need to examine the Y-values corresponding to the X-values given in the table.

Step 1: Identify the given values from the table. The pairs are as follows: - For $X = -1$ , $Y = 2$ - For $X = 0$ , $Y = 4$ - For $X = 1$ , $Y = 7$
Step 2: Check if all Y-values are the same. Compare Y-values for each X-value:

Y = 2

X = -1

Y = 4

X = 0

Y = 7

X = 1

Since the Y-values (2, 4, and 7) are not the same, the function is not constant.

Thus, the table does not represent a constant function. The correct choice is: No.

Answer

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Representing a Function Verbally and with Tables

Verbal representation of a function

Tabular representation of a function

Test yourself on representations of functions!

Example 1

Practice of verbal and tabular representation of a function

Example 2

Example 3

Example 4

Example 5

Y=X4 Y=\frac{X}{4} Y=4X​

Example 6

Example 7

Example 8

Y=X Y=\sqrt{X} Y=X​

Example 9

Y=X2−3 Y=\frac{X}{2}-3 Y=2X​−3

Example 10

Y=2X+5 Y=2X+5 Y=2X+5

Review questions

Examples with solutions for Representing a Function Verbally and with Tables

Exercise #1

Video Solution

Step-by-Step Solution

Answer

Exercise #2

Video Solution

Step-by-Step Solution

Answer

Exercise #3

Video Solution

Step-by-Step Solution

Answer

Exercise #4

Video Solution

Step-by-Step Solution

Answer

Exercise #5

Video Solution

Step-by-Step Solution

Answer

More Questions

Representations of Functions

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

$Y=\sqrt{X}$

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

$Y=2X+5$