Is $(f+g)x=f(x)+g(x)$ always true? My Attempt: 
If Variable is same then  $$(f+g)x=f(x)+g(x)$$  it's always true.
Thanks in Advance..
 A: The equation $(f+g)(x)=f(x)+g(x)$ is how addition of functions is generally defined (where “generally” here is more of a hedge word; I'm not aware of any other definition in use, however I don't want to completely exclude the possibility). It is called “pointwise addition”.
More exactly, if $A$ is an additive structure (that is, a set on which addition is defined), then given two functions $f:X\to A$ and $g:Y\to A$, then their sum is the function $(f+g):(X\cap Y)\to A$ with $(f+g)(x) = f(x)+g(x)$.
A: Whenever it makes sense to talk about $f + g$, it's true. Specifically, it makes sense when:


*

*The domains of $f$ and $g$ to be the same set $A$,

*The codomains of $f$ and $g$ to be the same set $B$,

*The operation $+$ makes sense on $B$.


By the third point, I mean that $B$ is a set like $\Bbb{R}$, or $\Bbb{C}$, or $M_{2 \times 2}(\Bbb{R})$, or just some other set with some sensible addition defined on it. All we require is that the set be closed under the $+$ operation, i.e. $b_1 + b_2 \in B$ for all $b_1, b_2 \in B$.
For example, if $f, g : \Bbb{C} \to \Bbb{C}$ such that $f(z) = \overline{z}$ and $g(z) = |z|$, then $f + g$ makes sense, and we define $f + g : \Bbb{C} \to \Bbb{C}$ by
$$(f + g)(z) = f(z) + g(z) = \overline{z} + |z|.$$
If, on the other hand, we had $f : \Bbb{R} \to \Bbb{R}$ and $g : M_{2 \times 2}(\Bbb{R}) \to \Bbb{R}$ defined by $f(x) = 2x$ and $g(M) = \operatorname{Tr}(M) - 6$, then $f + g$ would not make sense, since $f$ and $g$ have different domains. We could not say $(f + g)(x) = f(x) + g(x)$, because it's not clear whether $x$ is a real number, or a two-by-two real matrix.
So yeah, whenever $f + g$ makes sense, what you wrote is true.
