Is study of function important for higher mathematics?

Question

I just started the study of calculus and encounter functions. Please tell me if functions are important for higher mathematics. Is graphing of function important?

Please tell how to study the chapter, it really troubles me.

Answer 1

Functions are, in many ways, one of the basic and fundamental objects in higher mathematics. There are many branches where functions (or generalization of functions) are crucial objects.

Understanding the graph of a function is a really useful skill in higher mathematics (esp. in those areas that deal with calculus). Often when trying to understand a function we can look at its graph as a "picture" of a function. It tells you how it changes when you change the input/argument of the function and tells you about the points where the function behaves in special ways, like having a maximum/minima. It can also give you an understanding what values a function takes, whether it is increasing faster than another function, whether it is close to zero or not, etc. That is one reason that I would recommend that when you are reading about functions in your text book that you graph them yourself on some graph paper and/or use a graphing calculator or a website like http://www.wolframalpha.com/ that can graph functions. I would recommend graphing the functions that are given in the book, even if there already are graphs given, and "mess around" with the graphing software by changing the function slightly to see what that did to the graph of the function. For instance, if your textbook has the graph of the function $y = x^2$ and you see that it curves upward, increasing more and more, you might want to ask yourself, "Why does the function increase as $x$ gets larger? Why is $x^2 < 1$ when $0 < x < 1$? How would the function change if I made it instead $y = -x^2$ or $y = x^2 + 1$ or $y = x^2 + x$ or $y = x^{2.1}$?" Being able to graph multiple functions at the same time can be helpful in seeing the change in the function when you change it slightly. Getting some understanding for the relationship between the graphs of functions and the equation for the function can be a very useful skill. This intuition for a function based on its graph is often exploited in higher mathematics, like analysis classes, where sometimes when asked questions about a strange function on a homework exercise, it asks the student to graph the function. This helps the student understand the function and gives them intuition for being able to solve the rest of the problem.

Sometimes spending some time to understand the problem can go a long way and even save you time. For instance, if you spend time working through examples like the ones suggested in the previous paragraph and it helps you understand functions better so that, say, when you learn about a specific function like $y = 2^x$ it "clicks" and its properties make more sense than they would have otherwise had you not spent the time trying to understand functions, then not only have you spent your time studying wisely so that learning new material is easier and possibly faster, but you enjoy it more and can get a still deeper understanding of the topic.

A resource that I found very useful when learning some of the elementary topics in math was Khan Accademy's and other videos that you can find by searching for your topic on YouTube. For instance, Khan Academy has a playlist here about functions and their graphs: https://www.khanacademy.org/math/algebra-home/alg-functions

Understanding functions is crucial in basically all mathematics starting at algebra I in high school. So, getting familiar with them in the short run (in taking precalculus, which is chock full of functions) and in the long run if you are interested in pursuing basically any academic field or being an educated person. Functions and their graphs come up in many fields on science (like biology and chemistry, but especially physics and engineering) but also in everyday like in interpreting data given about history or provided by statistics or economics. That is why they often teach and ask questions about graphs in history classes, since graphs are visual ways to depict and understand data. Likewise, in more function-intense fields, understanding functions is very useful, so understanding graphs is very useful since understanding graphs is often a visual way of interpreting what a function "really means" and what it does.

That being said, yes, functions are important in higher mathematics. Because of that, graphs of functions are important. There are multiple ways of studying, but math is not a spectator sport. If you want to learn you have to get your hands dirty (with pencil lead or pen ink) and do the math. I would suggest taking notes while you read and trying and working out examples in the book yourself, before and after you read it. If you spend time trying to find the solution to an example before you read the answer and a solution, then you will get more out of it, even if you struggle with the problem and don't solve it by yourself. That being said, working through problems in the book is a very helpful thing to do. Try doing the ones that have the answers in the back of the book or try finding problems with answers online through videos that present and solve the problems or websites like http://tutorial.math.lamar.edu/Classes/Alg/Alg.aspx . Remember, working on a problem is not to just get to the right answer (which might be the overtone that is expressed in high school classes) but to understand the problem. Higher mathematics is all about understanding and asking questions. Of course, there are disciplines in higher mathematics that involve a lot of computation, especially the ones that have many applications to the sciences, but understanding how computations work and understanding in general are the big things in higher mathematics. In math it often takes work to understand what is going on, but it is a good thing to ask about how you can understand the material that you are learning. I hope that you got something out of this and do well. Finding out how you learn math best is a thing that you have to learn for yourself, since every person is different, but these are some things that have worked for me. Have a great day. :)