What is the intuitive way to understand addition of functions?

Question

I am studying some basic maths to refresh my knowledge. It is told that like numbers we can add two functions. While I got the mechanics of it but I don't understand the importance of it. Similar is the case for subtraction, division and multiplication.

Am I missing some insight that may help to get most out of mathematics?

Update 1

Thinking backwards I see that a simple function such as $x^2 + 2x + 5$ is made of three functions.

$f(x) = x^2$
$g(x) = 2x$
$f(x) = 5$

Still thinking the benefit of combining and breaking things like this in mathematics.

Oh, so you want to know the benefits of doing this? Is that your question? — Simply Beautiful Art, Mar 18 '16 at 19:19
@SimpleArt yes, more importantly how it would change my intuition. — vivek, Mar 19 '16 at 04:42
Well, perhaps you could start by rephrasing your question. And to know the benefits would make this question less vague and more precise for someone to answer it. — Simply Beautiful Art, Mar 19 '16 at 23:18

Eric Towers · Accepted Answer · 2016-03-18T18:42:54.510

Say I start a business. I have a function modelling revenue that says $f(\text{time spent}) = \text{revenue}$. This is a mapping from hours to money. It is also a machine that takes hours as input and produces amounts of money as output.

Now I've been in operation for a while and decide to hire an assistant to work in tandem with me. The assistant's contribution is modeled by $g(\text{time spent}) = \text{revenue}$. What is actually observed at the cash register is $f+g$, since I and the assistant always work the same amount of time.

score 1 · Answer 2 · answered Mar 18 '16 at 18:16

1

A function has an input and an output. To add two functions, you just add the two outputs resulting from the same input. If $f(60)=12$ and $g(60)=5$, then $(f+g)(60)$ is just $f(60)+g(60)$, i.e. $12+5$.

answered Mar 18 '16 at 18:16

Michael Hardy

1

I have understood function as nothing but a special kind of mapping. So, basically we are creating a new function from the two functions, but why or when we need to do that? any practical example? – vivek Mar 18 '16 at 18:19
@vivek Input$\to$output $=$ mapping. – Simply Beautiful Art Mar 18 '16 at 18:20
Why do we need to add two numbers at all? Numbers have algebraic properties which make them interesting. We would like to extend those interesting algebraic properties to functions. If you see intuitive reasons to talk about the sum of two numbers, well, the sum of two functions is also intuitively useful for the same reason. Not to mention the fact that being able to decompose a function as a sum of simpler functions is often very useful, such as in calculus when you take derivatives. – Vik78 Mar 18 '16 at 18:35

What is the intuitive way to understand addition of functions?

2 Answers2