1. ## What is the purpose of adding functions?

Hi!
I know that you can add functions together, like f(x) = 4x+3 and g(x) = 7x-5 and get a new function, h(x) = 11x-2
Why would someone want to add linear, or quadratic, or whatever functions together? Like are there real world applications for this?
I know you can add the derivatives of the functions together too, and you get h'(x) = 11. In what case would you want to add the derivatives together?

Thanks!

2. ## Re: What is the purpose of adding functions?

There are many applications that can be interpreted as a sum of two functions. If, for example, you are standing in an elevator going up at speed v relative to the ground. Your height above the ground is f(t)= vt. You drop a ball which then has distance below your hand g(t)= -gt^2/2. The height of the ball above the ground is f(t)+ g(t)= -gt^2/2+ vt.

That can also be used the other way. If I have a problem that involves, for whatever reason, the function h(x)= 3x^4+ sin(x) and want to differentiate it, rather than having to memorize a "rule" for differentiating a power of x plus a trig function, I can think of it as the sum of f(x)= 3x^4 and g(x)= sin(x) and differentiate each separately.

3. ## Re: What is the purpose of adding functions?

Consider the relatively simple function

$f(x) = x^4 - 3x + 2.$

How do you actually compute it its value for any value of x. You compute $x^4\ and\ then\ -\ 3x + 2$ and add them together.

In other words, you treat $f(x) = x^4 - 3x + 2\ as\ f(x) = g(x) + h(x),\ where\ g(x) = x^4\ and\ h(x) = - 3x + 2.$

Yes, you can conceptually view a function as a single entity, but it often is far more useful or practical to view it as being a combination of simpler functions.

4. ## Re: What is the purpose of adding functions?

Later thought

For a practical example of combining functions for a conceptual (rather than mechanical) purpose outside the realms of natural science, consider the profit function in economics:

$\pi(p,\ q) = \sigma(p,\ q) - \kappa(q),\ where$

$\sigma(p,\ q) = sales\ revenue$

$\kappa(q) = cost\ of\ production$

$q = quantity\ sold$

$p = price\ per\ unit\ sold$