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What is a Linear Function?

In Mathematics, the term linear function refers to two separate but interconnected concepts

In calculus and associated areas, a linear function is a polynomial function of degree 0 or 1, or is the 0 polynomial

It can also be called a linear map in linear algebra and functional analysis.

You will have to learn many basic mathematical functions over the course of your math study. The most basic and the foremost is the linear function. It is used to explain the relation in a straight line. This straight line can be at any angle or in any location. It can generally be described as y = m x + b.

We all face the realities of life in our day-to-day affairs. One of them is mathematical rules. Do we all know how much of the world runs by mathematical rules? One of them is linear function which is also known as linear system which can be described with a linear equation, as already mentioned in the above lines.

As one of the major mathematical tools, the linear system has multiple uses in our real lives. Life is full of events. The output of any system doubles if we double the input, at the same time, output cuts if we cut the input. This is how a linear system or linear function works. All linear systems can be described with linear equations.

Simply, a linear function is a function that graphs to a straight line. Mathematically, this means that if a function has one or two variables with no exponents (y=x) or powers. If the function has more variables (y=x+ pi), the variable must be constants or the known variables to remain the function of a linear function.

To identify the linear function(s) you can make a checklist of several conditions the function must fulfill.

  • 1-The first condition a linear function must satisfy is that it must have either one or two real variables. In case, another variable is present it must be a constant or known variable. For example, the function (C = 2 x pi x r) is a linear function because only the ‘C’ and ‘r’ are the real variables with the pi being a constant.
  • 2-The second condition a function needs to meet to be a linear is that none of the variables can have the power or exponent to them. They cannot be squared, cubed or anything else. All variables must be in numerator.
  • 3-The condition to fulfill is that the function must graph to a straight line. Any kind of arc or curve disqualifies the function to be a linear function.

So the linear function(s) are some kind of straight line when graphed. The plotted line could be in an upward and downward direction, left and right direction or slanted but the line needs to be always straight. It does not matter where on the graph the function is plotted as long as the line draws out straight.

If you comprehend that the function is linear, you can map the graph by using simply two points. If you are not sure, you can use three or four points to have a dual check

How a linear function acts in our daily lives?

Have you ever been in your kitchen to double your favorite pastry recipe? If so, you have put a linear function to work. Let’s see how; If one big pastry needs half a cup of butter, two cups of fine flour, three-fourth teaspoon of baking powder, some two or three eggs and one cup of sweetener and milk, the two big pastries will need one cup of butter, four cups of fine flour, one and a half teaspoon of baking powder, some four or six eggs and two cups of sweetener and milk. It means to get twice the output, you doubled the input. You might not have the idea that you just used the linear function by using a linear equation. But, believe it or not, that is exactly what you did.

Linear function in action – Snow Melting

Let us suppose that a water region wants to comprehend how much extra snow melting it can presume this year. The flux comes in from a large valley, and each year the region evaluates the melt and subsequent water supply. It gets 50 acre-feet from every 5 inches of melt. This year assessors measure 5 feet and 5 inches of snow. The region put that figures in the linear function by making the equation as (50 acre-feet/5 inches) * 65 inches. The officials at water region can expect 650 acre-feet of snow melting in shape of water.

Linear Function – Use it for fun

Julia wants to fill her swimming pool because it's springtime. But she, at the same time, does not want to stand there all the day. And she also does not want to waste water over the edges of her pool. She calculates that it takes thirty minutes to fill the pool level by four inches. She wants to fill her pool to a depth of four feet. She has forty-four more inches to fill. Julia calls in the linear function and figures out her linear equation as (30 minutes/4 inches) * 44 inches, so she comes to know that she has to wait for five hours and thirty more minutes to get her swimming pool filled up to four feet.

Linear Function – How often should you cut your lawn?

Smith has also noticed that it’s springtime. He also noticed that grass has been growing in his lawn at the rate of two inches in two weeks. He thinks that grass should not be taller than two and a half inches. At the same time, he does not like to cut it shorter than one and 3/4 inches. To determine how often he should cut the lawn, he employed linear function and put the figures in a linear expression: (days/grown inches) x desired inches equals to (14/2) x 3/4. This calculation tells him that he needs to cut his lawn every five and 1/4 days. Smith ignores the fraction (1/4) and cuts the lawn every five days.

Linear function is everywhere

It is not difficult to see the linear function in action in other similar situations. If you are going to throw a family bash and want to buy beer for your guests and got $50 in your pocket, a linear equation will tell you how much you can offer. In case if you want to know how much wood you need to burn overnight or desirous of calculating your remuneration or looking to redo up your stairs and bedrooms and how much paint you need or want to buy gasoline to and from your aunt’s house, linear function equations provide you with the solution. Hence, we can say that linear systems are everywhere.

Linear function: Actually they are not everywhere

One of the absurdities is that every linear system is also a nonlinear system. Thinking you can make one big pastry by multiplying a recipe without keeping consideration of the ratio of the ingredients will probably not work. If there is really a heavy snowfall than expected in a year and flakes get pushed up against the walls of the valley, the water district's estimate of attainable water will be off. After having the pool full and getting water washing over its edges, Julia cannot dig the pool any deeper. This simply means that all linear systems mostly have a linear regime (a section over which the linear system rules apply) and a nonlinear regime (where these rules do not). So you need to stay in the linear regime if you want the linear function equations to be held true.

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