How to make the Table of Values of a Linear Function?

In mathematics, a function defines a relationship between an independent variable and a dependent variable. In simple words, a function is a relationship between inputs and outputs in which each input is connected to exactly one output. If every element in set A has exactly one and only one image in set B, then the relation is said to be a function. Every function has a domain and a codomain, where a domain is a set of input values and a codomain, or range, is the set of possible output values for which the function is defined. The domain and codomain of a function are non-empty sets.  If there exists a function from A → B and (a, b) ∈ f, then f (a) = b, where "a" is the image of "b" under "f" and "b" is the preimage of "b" under "f" and set A is the domain of the function and set B is its co-domain.

Examples of a function

• The formula for the circumference/perimeter of a circle is P = 2πr, where r is the radius of a circle. We can say that circumference/perimeter (P) is dependent on the radius (r) of the circle. In the language of functions, we say that P is defined as a function of r.
• The area (A) of a square A is a function of its side length. The dependence of A on s is given by A = 4s².

Table Values of a Function

The table values of a function are referred to as the list of numbers that can be used to substitute for the given variable. By using this variable within the equation or in the other function, it is simple to determine the value of the other variable or the equation's missing integer. In the table of values of a function, there are two kinds of variables, namely an independent variable and a dependent variable. For any equation of a function, an independent variable is selected independently to determine the value of a dependent variable, which is the output of the given function. The table of values is unique for every function. A graph of the given function can be plotted easily after the determination of the values of the independent and dependent variables. There are many uses and applications for tables of values of a function. These are used in the fields of mathematics, physics, and engineering.

How to make the Table of Values of a Function?

A function is typically represented by f(x), where x is the input, and its general representation is y = f(x). 

1. Create the table first, then choose a range of input values.
2.  In the left-hand side column, substitute each input value into the given equation.
3. To determine the output value, evaluate the equation in the middle column. (A middle column is optional as the table of values just contains the input (independent variable) and output (dependent variable) pair.)
4. Now, note down the output values in the right-hand side column.

Let us solve an example to understand the concept better.

Example: Write the table of the value for the function y = √x.

Here, the input is x and the output is y, where y = √x.

x value	Equation y = √x	y value
0	y = √0 = 0	0
1	y = √1 = 1	1
4	y = √4 = 2	2
9	y = √9 = 3	3
16	y = √16 = 4	4
25	y = √25 = 5	5

Sample Problems

Problem 1: Write the table of values for the function y = 3x + 5.

Solution:

Here, the input is x and the output is y, where y = 3x + 5.

x value	Equation y = 3x +5	y value
-2	y = 3(-2) + 5 = -6 + 5 = -1	-1
-1	y = 3(-1) + 5 = -3 + 5 = 2	2
0	y = 3(0) + 5 = 0 + 5 = 5	5
1	y = 3(1) + 5 = 3 + 5 = 8	8
2	y = 3(2) + 5 = 6 + 5 = 11	11

Problem 2: Write the table of values for the function P = 4s, where P is the perimeter of a square and a is its side length.

Solution:

Here, the input is s and the output is P, where P = 4s.

s value	Equation P = 4s	P value
1	4 × 1 = 4	4
2	4 × 2 = 8	8
3	4 × 3 = 12	12
4	4 × 4 =16	16
5	4 × 5 = 20	20

Problem 3: Write the table of values for the function y = 2^x + 3^x.

Solution:

Here, the input is x and the output is y, where y = 2^x + 3^x .

x value	Equation y = 2x + 3x	y value
-2	y = 2-2 + 3-2 = 1/22 + 1/32 = 1/4 + 1/9 = 13/36 = 0.3611	0.3611
-1	y = 2-1 + 3-1 = 1/2 + 1/3 = 5/6 = 0.834	0.834
0	y = 20 + 30 = 1 + 1 = 2	2
1	y = 21 + 31 = 2 + 3 = 5	5
2	y = 22 + 32 = 4 + 9 = 13	13
3	y = 23 + 33 = 8 + 27 = 35	35

Problem 4: Write the table values for the function y = cos x × sin x.

Solution:

Here, the input is x and the output is y, where y = cos x × sin x.

x value	Equation y = cos x × sin x	y value
0°	y = cos 0 sin 0 = 1 × 0 = 0	0
30°	y = cos 30 sin 30 = √3/2 × 1/2 = 3/4	√3/4
45°	y = cos 45 sin 45 = 1/√2 × 1/√2 = 1/2	1/2
60°	y = cos 60 sin 60 = 1/2 × √3/2 = 3/4	√3/4
90°	y = cos 90 sin 90 = 0 × 1 = 0	0
180°	y = cos 180 sin 180 = -1 × 0 = 0	0

Problem 5: Write the table values for the function y = x² - 5x + 6.

Solution:

Here, the input is x and the output is y, where y = x² - 5x + 6.

x value	Equation y = x2 - 5x + 6	y value
-3	y = (-3)2 - 5(-3) + 6 = 9 + 15 + 6 = 30	30
-2	y = (-2)2 - 5(-2) + 6 = 4 + 10 + 6 = 20	20
-1	y = (-1)2 - 5(-1) + 6 = 1 + 5 + 6 = 12	12
0	y = 02 - 5(0) + 6 = 0 - 0 + 6 = 6	6
1	y = 12 - 5(1) + 6 = 1 - 5 + 6 = 2	2
2	y = 22 - 5(2) + 6 = 4 - 10 + 6 = 10- 10 = 0	0
3	y = 32 - 5(3) + 6 = 9 - 15 + 6 = 15 - 15 = 0	0