Question

Difference between (f+g)(x) and f(x) + g(x)?
I know that (f+g)(x) = f(x) + g(x).

I know that if we had f(x) = 2x + 5 and g(x) = 6x² + 3, then f(x) + g(x) = 6x² + 2x + 8, but I don't understand what (f+g)(x) is supposed to "mean".

Answer 1

it means just what you think it does. the point is that if you have a function f and a function, then you can create another function f+g

in order to define the function f + g you have to say what it does. so what is
\[(f+g)(x)\]?
well it is
\[(f+g)(x)=f(x)+g(x)\]

Answer 2

but don't get married to the variable "x" the function is the action, it has nothing to do with what you call the variable. by which i mean if say

\[S\] is the squaring function, then
\[S(x)=x^2,S(t)=t^2,S(z)=z^2,S(4)=4^2, S(t-1)=(t-1)^2 \] etc

Answer 3

so if you have two functions, they do not have to be functions of "x" you could have sine, cosines, log, exp, the square root function, the reciprocal function etc. the sooner you divorce yourself from thinking about
\[f(x)\] and start thinking about the function
\[f\] the better

Answer 4

probably more than you wanted to know, so here is a basic example.
Let
\[f(t)=\sqrt{t}, \text{for }t\geq 0\]
\[g(x)=\frac{1}{x}, \text{for }x\neq 0\] then
\[(f+g)(z)=\sqrt{z}+\frac{1}{z}, \text{for }z>0\]

Answer 5

ok, the last example made it much more clear :) hehe thanks!

Answer 6

that is what i figure.