A linear equation is a first degree equation in x and y. For example 2x - y = 6 is a linear equation. Graphically a linear equation is represented by a straight line in two dimensional coordinate system. The linear equation when written in the slope - intercept form y = mx + b can be regarded as a linear function f(x) = mx + b, for every input value of x the rule brings out exactly one output.

Any linear equation can thus represent a linear function and the equation can be used to interpret the function.

The slope intercept form of an equation can be regarded as a linear function rule. That is a linear function can be written in the form
f(x) = mx + b where m and b are constants.

For example, the linear equation y = 2x - 3 can be regarded as a linear function f(x) = 2x - 3. Note here the dependent variable can be replaced by the function notation f(x) and the expression 2x -3 is the rule to find the output for the function.

The constant 'm' in f(x) = mx + b gives the rate of change of the function and the 'b' represents the function value when x = 0. In other words f(0) = b.

For the function f(x) = 2x - 3, 2 is the rate of change and f(0) = - 3.

We discuss the significance of theses values and how we use them to write an equation to represent a linear function.
The graph of a linear function is a straight line. The steps for making the graph of a linear function is as follows:
  1. Write the equation in slope - intercept form, that is y = mx + b form.
  2. Make a table of values choosing convenient values for x and finding the corresponding y values using the equation got in step 1.
  3. Plot the ordered pairs. Join the points to get the straight line graph.

Solved Example

Question: Graph f(x) = 3x - 2
Solution:
 
Step 1:
The function rule is written as y = 3x - 2
Step 2: Table for the Graph
Two points are sufficient to draw a straight line. But in order to ensure the graph is done correct, we can find three points on the graph.
Let us choose x values as -1, 0 and 1. The corresponding y values are got using the equation
y = 3x - 2 as follows:
x = - 1  ⇒  y = 3(-1) - 2 = -3 - 2 = -5
x = 0    ⇒  y = 3(0) - 2 = 0 -2 = -2
x = 1    ⇒  y = 3(1) - 2 = 3 - 2 = 1
The table for the ordered pairs looks as follows:
  x 
 y = 3x - 2 
   (x, y) 
 -1         -5  
  (-1, -5)
  0        -2
  (0, -2)  
  1        1
  (1, 1 )

Step 3:
Plot the points. Join the points and extend the straight line on either side. The graph is shown below.

Linear Graph


 

The domain of a function is set of all input values (x values) for which the function is defined. A linear function f(x) = mx + b does not impose any restriction on the input variable and is defined for all values of x.

Hence the domain of a linear function is all real values. This can be written in Set builder form as { x | -$\infty$ < x < $\infty$ } or in interval form as (-$\infty$, $\infty$).

A linear function is a polynomial and in general the domain of any polynomial is the set of all real numbers.
The Range of a function is the set of all values the function can assume. You know a linear function is graphically represented by a straight line graph.

Straight lines can be extended at either end without bounds. This means the function or y value assumes all real values.
Hence the range of a linear function is also all real numbers or -$\infty$ < y < $\infty$.
All linear functions have one-to-one correspondence. Hence the inverse function can be found for any linear function.

Solved Example

Question: Find the inverse of f(x) = 4x - 7.
Solution:
 
y = 4x - 7                   Write in the form y = mx + b
x = 4y - 7                   Swap x and y.
x + 7 = 4y

⇒ y = $\frac{1}{4}$x + $\frac{7}{4}$     Equation solved for y

f-1 (x) = $\frac{1}{4}$x + $\frac{7}{4}$     Inverse of f(x)

It can be noted that the inverse of a linear function is also linear.
 

It may be required to write a linear function rule when the graph is given. We know a linear function is of the form f(x) = mx + b and we need to determine the values of m and b from the graph. We know in the linear equation y = mx + b, m represents the slope of the line and b the y intercept. The y intercept can be found from the graph by identifying the point where the line crosses the y axis.

How to find slope m from the graph of the linear function?
  1. Read the coordinates (x1, y1) and (x2, y2) of two points on the graph.
  2. Calculate the vertical change in the coordinates, rise = y2 - y1.
  3. Calculate the horizontal change in the coordinates, run = x2 - x1.
  4. The slope of the function is calculated as m = $\frac{rise}{ run}$.

Solved Example

Question: Find the Linear function for the graph given below.

Write Function


Solution:
 
The line cuts the y axis at the point (0, 5).  Hence the y intercept b = 5
Two points can be identified as (-4, 3) and ( 2, 6).
Rise = 6 - 3 = 3    and  Run = 2 - (-4) = 2 + 4 = 6

Hence Slope m = Rise / Run = $\frac{3}{6}$ = $\frac{1}{2}$

Thus the Linear function for the graph given is y = $\frac{1}{2}$x + 5.
 

Let us learn how to find the function rule for a linear function expressed in table form. Our task is to find the constants m and b for the linear model f(x) = mx + b. Let us consider the table showing the distance y Km traveled in time x hrs.

Time
x hours

Distance
y Km

2
142
4
274
6
406
8
538
10
670

We can observe that the change in successive x values and successive y values are constant. The difference between two consecutive x values always 2 and the difference between two y values is always 132. This suggests that the table values represent a linear function. The constant m which is viewed as the slope of the linear graph is present here as the constant rate of change.
m = Rate of change = $\frac{Change\ in\ dependent\ variable\ y}{ Change\ in\ the\ dependent\ variable\ x}$
= $\frac{132}{2}$ = 66

Now the linear function can be assumed as y = 66x + b.

To find the value of the constant b, we can use one of the ordered pairs from the table. Substituting x = 2 and y = 142 in the above model,
142 = 66(2) + b
132 + b = 142 ⇒ b = 10.
Hence the linear function that is represented by the table is, f(x) = 66x + 10 for 2 ≤ x ≤ 10.
In calculus, the derivative of a function f(x) is written as the function f ' (x) (Read as f prime x). f ' (x) gives the rule to find the slope of the tangent to the graph for a number x.

In the case of a linear function, the tangent at any point x coincides with the graph of f(x) itself. In other words, the derivative f ' (x) of a linear function f(x) is the constant representing the slope of the graph of f(x).

Example: f(x) = 3x + 8
f ' (x) = 3 which is the slope of the line y = 3x + 8.
Verify whether the given equation can represent a linear function. Explain your reasoning.

(a) $\frac{3}{4}$x + $\frac{1}{2}$y = 1 (b) x - xy + y = 3

Solution:
The equation (a) is a first degree equation in x and y and hence it represents a linear function. The equation can be rewritten as

$\frac{1}{2}$y = -$\frac{3}{4}$x + 1 Separate variables

2 . $\frac{1}{2}$y = (-$\frac{3}{4}$x + 1) . 2 (Multiplying by 2)

y = $\frac{-3}{2}$x + 2

f(x) = $\frac{-3}{2}$x + 2 . Linear function of the form f(x) = mx + b.

The equation (b) contains a second degree term xy. Thus the equation cannot be written in the form y = mx + b and hence does not represent a linear function.

Example: Check whether the data given in the table represents a linear function. If yes, find the function rule that defines the table.
x
f(x)
Difference in f(x)
4 73 80
8 153 80
12 233 80
16 313 80
20 393 80
24 473 80

It can be noted that for a constant change of 4 in x values, f(x) increases by a constant value of 80. Hence the table represents a linear function.

m = Rate of change = $\frac{Change\ in\ f(x)}{Change\ in\ x}$ = $\frac{80}{4}$ = 20

Hence f(x) is of the form, f(x) = 20x + b.

Here (4, 73) is an ordered pair that satisfies the equation.

=> f(4) = 20(4) + b = 73 ⇒ b = 73 - 80 = -7.

Hence the rule that defines the table is f(x) = 20x - 7.

Solved Examples

Question 1: Clay works as a part time Bar tender for 4 hrs a day. His earnings per day included a fixed amount of 20 dollars for 4 hrs and a 2% commission on the sales attended.
    (a) Write a linear function to calculate his earnings per day for a sales attended worth x dollars.
    (b) If he had attended a sales of 900 dollars during his shift, calculate his earning for the day.
Solution:
 
Let the function model for Clay's daily earnings be E(x) = mx + b for a sales worth x dollars. While the constant b in the model indicates his fixed earning for a working day, m represents the commission percent given to him expressed as a decimal.
   Hence b = 20 and m = 0.02.
   Thus the model to be used for calculating Clay's earnings per day is E(x) = 0.02x + 20
   When the sales amounts to 900 dollars, his earnings for the day = 0.02(900) + 20 = 18 + 20 = 38 dollars.

 

Question 2: Maria has a contract to decorate a Ball room with Tulips. The Ball room has to be decorated with 50 Bouquets which costs a total of 600 dollars. Additionally each dancer is given flowers worth 12 dollars.
    (a) Write a linear model for cost of decoration if the Ball consisted of x dancers.
    (b) Find the total cost of decoration if 80 dancers participated in the dance.
Solution:
 
For the situation described above, the fixed cost 600 dollars corresponds to b in the linear model y = mx + b. m corresponds to the cost of flowers given to each dancer.
    Hence the cost function for the Ball Room decoration  C(x) = 12x + 600  where x represents the number of dancers.
    If 80 dancers participated in the dance, the cost of decoration = 12 (80) + 600 = 960 + 600 = 1560 dollars.