## Linear Function Formula

Before going to learn the linear function formulas, let us recall what is a linear equation and what is a function. A linear equation is an equation in which every term is either just a constant or the product of a constant and a variable of exponent 1. A function is a relation in which every input has exactly one output. Every linear equation represents a line. All lines except vertical lines are functions. Let us learn linear function formulas along with a few solved examples.

## What Are Linear Function Formulas?

There are multiple linear function formulas to find the equation of a line depending on the available information. Linear function formulas are applicable to find the equation of any line except vertical line as vertical line is NOT a function. The linear function formulas are:

A, B, and C are constants.

## Solved Examples Using Linear Function Formulas

**Example 2 **: A car rental in a city rents its cars for an upfront fee of $5 and a daily fee of $30 based on the number of days the car was rented. If x is the number of days and y is the total fee then write the linear relationship between x and y.

**Solution:**

The upfront fee = $5.

Daily fee = $30.

The total number of days = x.

So the fee for x days = 30x.

Using one of the linear function formulas (slope-intercept formula),

y = mx + b

y = 30x + 5.

**Answer: The required linear relationship is y = 30x + 5.**

## Linear Function

A linear function is a function that represents a straight line on the coordinate plane. For example, y = 3x – 2 represents a straight line on a coordinate plane and hence it represents a linear function. Since y can be replaced with f(x), this function can be written as f(x) = 3x – 2.

In this article, we are going to learn the definition of a linear function along with its graph, domain, and range. Also, we will learn how to identify a linear function and how to find its inverse.

## What is a Linear Function?

A **linear function** is of the form f(x) = mx + b where ‘m’ and ‘b’ are real numbers. Isn’t it looking like the slope-intercept form of a line which is given by y = mx + b? Yes, this is because a linear function represents a line. i.e., its graph is a line. Here,

- ‘m’ is the slope of the line
- ‘b’ is the y-intercept of the line
- ‘x’ is the independent variable
- ‘y’ (or f(x)) is the dependent variable

A linear function is an algebraic function.

### Linear Function Equation and Examples

The parent linear function is f(x) = x, which is a line passing through the origin. In general, a linear function equation is f(x) = mx + b and here are some examples.

- f(x) = 3x – 2
- f(x) = -5x – 0.5
- f(x) = 3

### Real Life Example of Linear Function

Here are some real-life applications of the linear function.

- A movie streaming service charges a monthly fee of $4.50 and an additional fee of $0.35 for every movie downloaded. Then the total monthly fee is represented by the linear function f(x) = 0.35x + 4.50, where x is the number of movies downloaded in a month.
- A t-shirt company charges a one-time fee of $50 and $7 per T-shirt to print logos on T-shirts. Then the total fee is given by the linear function f(x) = 7x + 50, where x is the number of T-shirts.
- The linear function is used to represent an objective function in linear programming problems, to help minimize the close, or maximize the profits.

### How to Find a Linear Function?

We use the slope-intercept form or the point-slope form to find a linear function. The process of finding a linear function is as same as the process of finding the equation of a line and is explained with an example.

**Example:** Find the linear function that has two points (-1, 15) and (2, 27) on it.

**Solution:**

The given points are (x₁, y₁) = (-1, 15) and (x₂, y₂) = (2, 27).

**Step 1:** Find the slope of the function using the slope formula:

m = (y₂ – y₁) / (x₂ – x₁) = (27 – 15) / (2 – (-1)) = 12/3 = 4.

**Step 2:** Find the equation of linear function using the point slope form.

y – y₁ = m (x – x₁)

y – 15 = 4 (x – (-1))

y – 15 = 4 (x + 1)

y – 15 = 4x + 4

y = 4x + 19

Therefore, the equation of the linear function is, f(x) = 4x + 19.

## Identifying a Linear Function

If the information about a function is given as a graph, then it is linear if the graph is a line. If the information about the function is given in the algebraic form, then it is linear if it is of the form f(x) = mx + b. But to see whether the given data in a table format represents a linear function:

- Compute the differences in x-values.
- Compute the differences in y-values
- Check whether the ratio of the difference in y-values to the difference in x-values is always constant.

**Example:** Determine whether the following data from the following table represents a linear function.

x | y |
---|---|

3 | 15 |

5 | 23 |

7 | 31 |

11 | 47 |

13 | 55 |

**Solution:**

We will compute the differences in x-values, differences in y values, and the ratio (difference in y)/(difference in x) every time and see whether this ratio is a constant.

Since all numbers in the last column are equal to a constant, the data in the given table represents a linear function.

## Graphing a Linear Function

We know that to graph a line, we just need any two points on it. If we find two points, then we can just join them by a line and extend it on both sides. The graph of a linear function f(x) = mx + b is

- an increasing line when m > 0
- a decreasing line when m < 0
- a horizontal line when m = 0

There are two ways to graph a linear function.

- By finding two points on it
- By using its slope and y-intercept

### Graphing a Linear Function by Finding Two Points

To find any two points on a linear function (line) f(x) = mx + b, we just assume some random values for ‘x’ and substitute these values in the function to find the corresponding values for y. The process is explained with an example where we are going to graph the function f(x) = 3x + 5.

**Step 1:**Find two points on the line by taking some random values

We will assume that x = -1 and x = 0.**Step 2:**Substitute each of these values in the function to find the corresponding y-values.

Here is the table of the linear function y = 3x + 5.

x | y |
---|---|

-1 | 3(-1)+5 = 2 |

0 | 3(0)+5 = 5 |

Therefore, two points on the line are (-1, 2) and (0, 5). **Step 3: **Plot the points on the graph and join them by a line. Also, extend the line on both sides.

### Graphing a Linear Function Using Slope and y-Intercept

To graph a linear function, f(x) = mx + b, we can use its slope ‘m’ and the y-intercept ‘b’. The process is explained again by graphing the same linear function f(x) = 3x + 5. Its slope is, m = 3 and its y-intercept is (0, b) = (0, 5).

**Step 1:**Plot the y-intercept (0, b).

Here, we plot the point (0, 5).**Step 2:**Write the slope as the fraction rise/run and identify the “rise” and the “run”.

Here, the slope = 3 = 3/1 = rise/run.

So rise = 3 and run = 1.**Step 3:**Rise the y-intercept vertically by “rise” and then run horizontally by “run”. This results in a new point.

(Note that if “rise” is positive, we go up and if “rise” is negative, we go down. Also, if “run” is positive”, we go right and if “run” is negative, we go left.)

Here, we go up by 3 units from the y-intercept and there by go right by 1 unit.**Step 4:**Join the points from**Step 1**and**Step 2**by a line and extend the line on both sides.

## Domain and Range of Linear Function

The domain of a linear function is the set of all real numbers, and the range of a linear function is also the set of all real numbers. The following figure shows f(x) = 2x + 3 and g(x) = 4 −x plotted on the same axes.

Note that both functions take on real values for all values of x, which means that the domain of each function is the set of all real numbers (R). Look along the x-axis to confirm this. For every value of *x*, we have a point on the graph.

Also, the output for each function ranges continuously from negative infinity to positive infinity, which means that the range of either function is also R. This can be confirmed by looking along the y-axis, which clearly shows that there is a point on each graph for every y-value. Thus, when the slope m ≠ 0,

**The domain of a linear function = R****The range of a linear function = R**

**Note:**

(i) The domain and range of a linear function is R as long as the problem has not mentioned any specific domain or range.

(ii) When the slope, m = 0, then the linear function f(x) = b is a horizontal line and in this case, the domain = R and the range = {b}.

## Inverse of a Linear Function

The inverse of a linear function f(x) = ax + b is represented by a function f^{-1}(x) such that f(f^{-1}(x)) = f^{-1}(f(x)) = x. The process to find the inverse of a linear function is explained through an example where we are going to find the inverse of a function f(x) = 3x + 5.

**Step 1:**Write y instead of f(x).

Then the above equation becomes y = 3x + 5.**Step 2:**Interchange the variables x and y.

Then we get x = 3y + 5.**Step 3:**Solve the above equation for y.

x – 5 = 3y

y = (x – 5)/3**Step 4:**Replace y by f^{-1}(x) and it is the inverse function of f(x).

f^{-1}(x) = (x – 5)/3

Note that f(x) and f^{-1}(x) are always symmetric with respect to the line y = x. Let us plot the linear function f(x) = 3x + 5 and its inverse f^{-1}(x) = (x – 5)/3 and see whether they are symmetric about y = x. Also, when (x, y) lies on f(x), then (y, x) lies on f^{-1}(x). For example, in the following graph, (-1, 2) lies on f(x) whereas (2, -1) lies on f^{-1}(x).

## Piecewise Linear Function

Sometimes the linear function may not be defined uniformly throughout its domain. It may be defined in two or more ways as its domain is split into two or more parts. In such cases, it is called a **piecewise linear function. **Here is an example.

**Example: **Plot the graph of the following piecewise linear function.

**Solution:**

This piece-wise function is linear in both the indicated parts of its domain. Let us find the endpoints of the line in each case.

**When x ∈ [-2, 1):**

x | y |
---|---|

-2 | -2 + 2 = 0 |

1 (hole in this case as 1 ∉ [-2, 1) ) | 1 + 2 = 3 |

**When x ∈ [1, 2]:**

x | y |
---|---|

1 | 2(1) – 3 = -1 |

2 | 2(2) – 3 = 1 |

The corresponding graph is shown below:

**Important Notes on Linear Functions:**

- A linear function is of the form f(x) = mx + b and hence its graph is a line.
- A linear function f(x) = mx + b is a horizontal line when its slope is 0 and in this case, it is known as a constant function.
- The domain and range of a linear function f(x) = ax + b is R (all real numbers) whereas the range of a constant function f(x) = b is {b}.
- These linear function are useful to represent the objective function in linear programming.
- A constant function has no inverse as it is NOT one-one.
- Two linear functions are parallel if their slopes are equal.
- Two linear functions are perpendicular if the product of their slopes is -1.
- A vertical line is NOT a linear function as it fails the vertical line test.

## Examples on Linear Functions

**Example 1:** The relationship between Celsius degrees and Fahrenheit degrees is linear. Some equivalent values are shown in the table below. Find the linear function representing the given data.

Celcius (°C) | Fahrenheit (°F) |
---|---|

5 | 41 |

10 | 50 |

15 | 59 |

20 | 68 |

**Solution:**

To find the linear function, it is sufficient to consider any two points from the table.

Let (x₁, y₁) = (5, 41) and (x₂, y₂) = (10, 50).

The slope, m = (y₂ – y₁) / (x₂ – x₁) = (50 – 41) / (10 – 5) = 9/5.

Uing the point-slope form,

y – y₁ = m (x – x₁)

y – 41 = (9/5) (x – 5)

y – 41 = (9/5) x – 9

y = (9/5) x + 32

From the table, the independent variable is C and the dependent variable is F. So the linear relationship is, F = (9/5) C + 32.

**Answer:** The linear relationship between the Celcius and Fahrenheit is, F = (9/5) C + 32.

**Example 2:** The cost (in dollars) of renting a car is represented by C(x) = 30 x + 20, where x is the number of days the car is rented for. Then what is the cost of renting the car for 10 days?

**Solution:**

To find the cost of renting the car for 10 days, substitute x = 10 in the given linear function.

C(10) = 30(10) + 20 = 300 + 20 = 320

**Answer:** The cost of renting the car for 10 days = $320.

**Example 3:** Considering the scenario of **Example 2**, if Ryan paid the total rent to be $470, then for how many days did he rent the car?

**Solution:**

The given linear function is C(x) = 470, where ‘x’ is the number of days that car is rented for.

470 = 30x + 20

450 = 30x

x = 450/30 = 15

**Answer:** Ryan rented the car for 15 days.

## FAQs on Linear Function

### Define Linear Function.

A linear function is a function whose graph is a line. Thus, it is of the form f(x) = ax + b where ‘a’ and ‘b’ are real numbers.

### What is the Formula to Find a Linear Function?

Since a linear function represents a line, all formulas used to find the equation of a line can be used to find the equation of a linear function. Thus, the linear function formulas are:

- Standard form: ax + by + c = 0
- Slope-intercept form: y = mx + b
- Point-slope form: y – y₁ = m (x – x₁)
- Intercept form: x/a + y/b = 1

Note that y can be replaced with f(x) in all these formulas.

### What is Linear Function Table?

Sometimes, the data representing a linear function is given in the form of a table with two columns where the first column gives the data of the independent variable and the second column gives the corresponding data of the dependent variable. This is called the linear function table.

### How to Graph a Linear Function?

To graph a linear function, find any two points on it by assuming some random numbers either for the dependent or for the independent variable and find the corresponding values of the other variable. Just plot those two points and join them by a line by extending the line on both sides.

### What is the Domain and Range of a Linear Function?

The domain and range of a linear function f(x) = ax + b where a ≠ 0 is the set of all real numbers. If a = 0, the domain is still the set of all real numbers but the range is the set {b}. Sometimes, the domain and range in a problem may be restricted to some interval.

### What is Linear Function Equation?

The linear function equation is nothing but the slope-intercept form. Thus, it is given by f(x) = mx + b where m is the slope and b is the y-intercept of the line.

### What are Linear Function Examples?

f(x) = 2x + 3, f(x) = (1/5) x – 7 are some examples of linear function. For real life examples of a linear function

### How do You Determine a Linear Function?

We can determine a linear function in the following ways.

- If the equation of a function is given, then it is linear if it is of the form f(x) = ax + b.
- If the graph of a function is given, then it is linear if it represents a line.
- If a table of values representing a function is given, then it is linear if the ratio of the difference in y-values to the difference in x-values is always a constant.

## Linear Equations

A **linear** equation is an equation for a straight **line**

### These are all linear equations:

## Different Forms

There are many ways of writing linear equations, but they usually have **constants** (like “2” or “c”) and must have simple **variables** (like “x” or “y”).

### Examples: These are linear equations:

## Slope-Intercept Form

The most common form is the slope-intercept equation of a straight line:

## Point-Slope Form

Another common one is the Point-Slope Form of the equation of a straight line:

## General Form

And there is also the General Form of the equation of a straight line:

Ax + By + C = 0 |

(A and B cannot both be 0) |

Example: 3x + 2y − 4 = 0

It is in the form **Ax + By + C = 0** where:

- A = 3
- B = 2
- C = −4

There are other, less common forms as well.

## As a Function

Sometimes a linear equation is written as a function, with f(x) instead of y:

y = 2x − 3 |

f(x) = 2x − 3 |

These are the same! |

And functions are not always written using f(x):

y = 2x − 3 |

w(u) = 2u − 3 |

h(z) = 2z − 3 |

These are also the same! |

## The Identity Function

There is a special linear function called the “Identity Function”:

f(x) = x

And here is its graph:

## Constant Functions

Another special type of linear function is the Constant Function … it is a horizontal line:

No matter what value of “x”, f(x) is always equal to some constant value.

## What is a Linear Function?

A **linear function** is a function which forms a straight line in a graph. It is generally a polynomial function whose degree is utmost 1 or 0. Although the linear functions are also represented in terms of calculus as well as linear algebra. The only difference is the function notation. Knowing an ordered pair written in function notation is necessary too. f(a) is called a function, where a is an independent variable in which the function is dependent. Linear Function Graph has a straight line whose expression or formula is given by;

** y = f(x) = px + q **

It has one independent and one dependent variable. The independent variable is x and the dependent one is y. P is the constant term or the y-intercept and is also the value of the dependent variable. When x = 0, q is the coefficient of the independent variable known as slope which gives the rate of change of the dependent variable.

**What is a Nonlinear Function?**

A function which is not linear is called nonlinear function. In other words, a function which does not form a straight line in a graph. The examples of such functions are exponential function, parabolic function, inverse functions, quadratic function, etc. All these functions do not satisfy the linear equation y = m x + c. The expression for all these functions is different.

## Linear Function Graph

Graphing a linear equation involves three simple steps:

- Firstly, we need to find the two points which satisfy the equation, y = px+q.
- Now plot these points in the graph or X-Y plane.
- Join the two points in the plane with the help of a straight line.

## Linear Function Table

See the below table where the notation of the ordered pair is generalised in normal form and function form.

A normal ordered pair | A function notation ordered pair |

(a,b) = (2,5) | f(a) = y coordinate, a=2 and y = 5, f(2) = 5 |

Using the table, we can verify the linear function, by examining the values of x and y. For the linear function, the rate of change of y with respect the variable x remains constant. Then, the rate of change is called the slope.

Let us consider the given table,

x | y |

0 | 3 |

1 | 4 |

2 | 5 |

3 | 6 |

4 | 7 |

Form the table, it is observed that, the rate of change between x and y is 3. This can be written using the linear function y= x+3.

## Linear Function Formula

The expression for the linear function is the formula to graph a straight line. The expression for the linear equation is;

**y = mx + c**

where m is the slope, c is the intercept and (x,y) are the coordinates. This formula is also called **slope formula**.

While in terms of function, we can express the above expression as;

f(x) = a x + b, where x is the independent variable.

### Linear Function Characteristics

Let’s move on to see how we can use function notation to graph 2 points on the grid.

**Relation**: It is a group of ordered pairs.**Variable**: A symbol that shows a quantity in a math expression.**Linear function**: If each term is either a constant or It is the product of a constant and also (the first power of) a single variable, then it is called as an algebraic equation.**Function**: A function is a relation between a set of inputs and a set of permissible outputs. It has a property that each input is related to exactly one output.**Steepness**: The rate at which a function deviates from a reference**Direction**: Increasing, decreasing, horizontal or vertical.

### Linear Function Examples

Graphing of linear functions needs to learn linear equations in two variables.

**Example 1** :

Let’s draw a graph for the following function:

F(2) = -4 and f(5) = -3

**Solution:**

- Let’s rewrite it as ordered pairs(two of them).
- f(2) =-4 and f(5) = -3

(2, -4) (5, -3)

How to evaluate the slope of a linear Function?

Let’s learn it with an example:

**Example 2 **:

Find the slope of a graph for the following function.

f(3) = -1 and f(-8) = -6

**Solution:**

Let’s write it again as ordered pairs

f(3) =-1 and f(8) = -6

(3, -1) (8, -6)

we will use the slope formula to evaluate the slope

(3, -1) (8, -6)

(x_{1 }, y_{1}) (x_{2 }, y_{2})

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