what does it mean to say that a function is linear
Linear Function
A linear function is a function that represents a straight line on the coordinate aeroplane. For case, y = 3x - ii represents a straight line on a coordinate plane and hence it represents a linear function. Since y tin can be replaced with f(x), this function can be written as f(x) = 3x - 2.
In this article, nosotros are going to learn the definition of a linear part along with its graph, domain, and range. We will too learn how to identify a linear role and how to notice its inverse.
| ane. | What is a Linear Function? |
| 2. | Identifying a Linear Function |
| 3. | Graphing a Linear Function |
| 4. | Domain and Range of Linear Function |
| five. | Inverse of a Linear Function |
| 6. | Piecewise Linear Function |
| vii. | FAQs on Linear Part |
What is a Linear Function?
A linear function is of the form f(10) = mx + b where 'm' and 'b' are real numbers. Isn't information technology looking similar the slope-intercept form of a line which is expressed as y = mx + b? Yeah, this is because a linear part represents a line, i.due east., 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(10)) is the dependent variable
A linear function is an algebraic function.
Linear Part Equation and Examples
The parent linear part is f(x) = 10, which is a line passing through the origin. In full general, a linear part equation is f(ten) = mx + b and here are some examples.
- f(x) = 3x - 2
- f(ten) = -5x - 0.five
- f(10) = three
Existent Life Instance of Linear Function
Here are some real-life applications of the linear role.
- A picture streaming service charges a monthly fee of $4.fifty and an boosted fee of $0.35 for every picture downloaded. Now, the total monthly fee is represented by the linear role f(x) = 0.35x + 4.50, where ten is the number of movies downloaded in a month.
- A t-shirt company charges a ane-fourth dimension fee of $50 and $seven per T-shirt to impress logos on T-shirts. So, the full fee is expressed 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 part in linear programming problems, to help minimize the close, or maximize the profits.
How to Find a Linear Role?
We use the gradient-intercept form or the point-gradient form to find a linear part. The procedure of finding a linear office is the aforementioned 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, xv) and (2, 27) on information technology.
Solution:
The given points are (x₁, y₁) = (-i, 15) and (x₂, y₂) = (two, 27).
Pace 1: Find the slope of the function using the slope formula:
thousand = (y₂ - y₁) / (x₂ - x₁) = (27 - xv) / (two - (-1)) = 12/3 = 4.
Step ii: Observe the equation of linear function using the point slope course.
y - y₁ = m (x - x₁)
y - 15 = 4 (10 - (-1))
y - fifteen = iv (x + one)
y - 15 = 4x + 4
y = 4x + nineteen
Therefore, the equation of the linear role is, f(x) = 4x + xix.
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 data about the part is given in the algebraic form, so it is linear if it is of the class f(ten) = mx + b. But to see whether the given data in a table format represents a linear office:
- Compute the differences in ten-values.
- Compute the differences in y-values
- Bank check whether the ratio of the difference in y-values to the difference in x-values is e'er constant.
Example: Determine whether the following data from the following table represents a linear function.
| x | y |
|---|---|
| 3 | fifteen |
| five | 23 |
| 7 | 31 |
| 11 | 47 |
| xiii | 55 |
Solution:
Nosotros volition 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 concluding column are equal to a abiding, the data in the given tabular array represents a linear part.
Graphing a Linear Function
We know that to graph a line, we but need any two points on it. If we find two points, so we can just join them by a line and extend it on both sides. The graph of a linear part f(x) = mx + b is
- an increasing line when m > 0
- a decreasing line when grand < 0
- a horizontal line when k = 0
There are two ways to graph a linear function.
- Past finding two points on information technology.
- By using its slope and y-intercept.
Graphing a Linear Function by Finding 2 Points
To find any two points on a linear function (line) f(x) = mx + b, we only assume some random values for 'ten' and substitute these values in the office to notice the respective values for y. The process is explained with an example where we are going to graph the function f(x) = 3x + five.
- Step 1: Find two points on the line by taking some random values
We will assume that x = -i and ten = 0. - Step ii: Substitute each of these values in the function to find the respective y-values.
Hither is the table of the linear function y = 3x + 5.
Therefore, two points on the line are (-1, ii) and (0, five).x y -1 3(-1)+5 = 2 0 3(0)+five = 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 Office Using Slope and y-Intercept
To graph a linear function, f(10) = mx + b, we can utilize its slope 'grand' and the y-intercept 'b'. The procedure is explained again past graphing the same linear function f(x) = 3x + 5. Its slope is, thousand = 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 gradient every bit the fraction rising/run and identify the "ascension" and the "run".
Hither, the gradient = 3 = 3/1 = rise/run.
So rise = three and run = 1. - Step 3: Rise the y-intercept vertically past "rise" and and then run horizontally by "run". This results in a new point.
(Note that if "rise" is positive, we go upwards and if "ascension" is negative, we go down. Too, if "run" is positive", we get right and if "run" is negative, nosotros get left.)
Here, we go up by three units from the y-intercept and thereby go correct by ane unit. - Stride iv: Bring together the points from Footstep 1 and Step 2 past a line and extend the line on both sides.
Domain and Range of Linear Part
The domain of a linear function is the ready of all existent numbers, and the range of a linear function is besides the set up of all real numbers. The following figure shows f(x) = 2x + 3 and g(x) = iv −x plotted on the same axes.
Annotation that both functions take on existent values for all values of x, which means that the domain of each function is the prepare of all real numbers (R). Look forth the 10-axis to confirm this. For every value of x, we take a signal on the graph.
Besides, the output for each part ranges continuously from negative infinity to positive infinity, which ways that the range of either function is too R. This can be confirmed past looking along the y-axis, which clearly shows that there is a point on each graph for every y-value. Thus, when the slope thousand ≠ 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.
(2) When the gradient, thousand = 0, then the linear function f(x) = b is a horizontal line and in this instance, the domain = R and the range = {b}.
Inverse of a Linear Function
The inverse of a linear office f(x) = ax + b is represented by a role f-i(10) such that f(f-ane(10)) = f-1(f(x)) = 10. The procedure to notice the inverse of a linear part is explained through an example where nosotros are going to find the inverse of a function f(x) = 3x + 5.
- Pace ane: Write y instead of f(10).
So the above equation becomes y = 3x + 5. - Step ii: Interchange the variables ten and y.
Then nosotros go ten = 3y + 5. - Pace 3: Solve the above equation for y.
10 - five = 3y
y = (ten - v)/three - Step four: Replace y past f-ane(x) and it is the inverse office of f(ten).
f-ane(x) = (x - five)/3
Notation that f(x) and f-1(ten) are always symmetric with respect to the line y = ten. Let us plot the linear part f(x) = 3x + 5 and its inverse f-1(x) = (10 - five)/3 and encounter whether they are symmetric well-nigh y = x. Likewise, when (10, y) lies on f(x), so (y, ten) lies on f-1(x). For case, in the following graph, (-i, 2) lies on f(x) whereas (2, -one) lies on f-one(ten).
Piecewise Linear Part
Sometimes the linear role may not be defined uniformly throughout its domain. It may be divers in two or more ways as its domain is split into two or more than parts. In such cases, it is called a piecewise linear function. Here is an instance.
Case: Plot the graph of the post-obit piecewise linear role.
\(f(x)=\left\{\begin{array}{ll}
x+ii, & x \in[-two,1) \\
2 x-iii, & x \in[1,ii]
\end{array}\correct.\)
Solution:
This piece-wise function is linear in both the indicated parts of its domain. Let the states discover the endpoints of the line in each instance.
When 10 ∈ [-2, 1):
| x | y |
|---|---|
| -2 | -ii + 2 = 0 |
| 1 (hole in this instance as 1 ∉ [-2, 1) ) | 1 + ii = 3 |
When x ∈ [i, 2]:
| ten | y |
|---|---|
| 1 | two(1) - 3 = -ane |
| two | ii(two) - three = 1 |
The corresponding graph is shown beneath:
Important Notes on Linear Functions:
- A linear part is of the form f(x) = mx + b and hence its graph is a line.
- A linear office f(ten) = mx + b is a horizontal line when its gradient is 0 and in this case, it is known as a constant role.
- The domain and range of a linear function f(ten) = ax + b is R (all real numbers) whereas the range of a constant function f(ten) = b is {b}.
- These linear function are useful to represent the objective part in linear programming.
- A abiding office has no inverse as it is Not a one-one function.
- Two linear functions are parallel if their slopes are equal.
- Two linear functions are perpendicular if the product of their slopes is -i.
- A vertical line is NOT a linear function as information technology fails the vertical line test.
☛ Related Topics:
- Linear Function Calculator
- Linear Function Formula
- Quadratic Function
- Odd Function
- Graphing Functions
- Changed Functions
Examples on Linear Functions
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Exercise Questions on Linear Functions
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FAQs on Linear Function
What is a Linear Function?
A linear function is a function whose graph is a line. Thus, information technology 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 tin exist used to find the equation of a linear function. Thus, the linear function formulas are:
- Standard form: ax + by + c = 0
- Slope-intercept course: y = mx + b
- Point-slope form: y - y₁ = m (x - x₁)
- Intercept grade: x/a + y/b = 1
Note that y can exist replaced with f(10) in all these formulas.
What is Linear Function Tabular array?
Sometimes, the data representing a linear role is given in the course of a table with two columns where the outset column gives the information of the contained variable and the second cavalcade gives the corresponding data of the dependent variable. This is called the linear function tabular array.
How to Graph a Linear Function?
To graph a linear office, detect whatsoever two points on information technology 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 bring together them by a line by extending the line on both sides.
What is the Domain and Range of a Linear Office?
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 however the fix of all real numbers but the range is the fix {b}. Sometimes, the domain and range in a trouble may be restricted to some interval.
What is a Linear Office Equation?
The linear role equation is the slope-intercept form. Thus, it is expressed as f(x) = mx + b where yard is the slope and b is the y-intercept of the line.
What are Linear Role Examples?
f(x) = 2x + 3, f(ten) = (1/v) x - 7 are some examples of linear function. For real life examples of a linear role, click here.
How to Determine a Linear Part?
We can determine a linear function in the following ways.
- If the equation of a office is given, and so it is linear if it is of the form f(ten) = ax + b.
- If the graph of a function is given, then it is linear if information technology 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 ten-values is always a abiding.
Source: https://www.cuemath.com/calculus/linear-functions/
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