Linear Transformations

"Show the following transformation is a linear transformation:
T :Map(R,R)→Map(R,R),
T(f(x))=f(x+1)"

I have tried but seem to be struggling:
T(f(X1)+f(X2))=T(f(X1+X2))=f(X1+X2+1)?
T(f(aX1))=f(aX1+1)?

Is this correct? I haven't done seen a proof of a linear transformation before so I'm unsure when doing the addition whether it is
T(f(X1+X2)) or T(f(X1)+f(X2)) or whether these are equal or when doing scalar whether it is
T(aF(X)) or T(F(aX1)) or whether these are also equal?

When showing that something is a linear transformation, you want $T(\lambda f(x) + \mu g(x)) = \lambda T(f(x)) + \mu T(g(x))$.

In particular here, you want to show that $T(\lambda f(x) + \mu g(x)) = \lambda f(x+1) + \mu g(x+1)$

I think it may be easiest if you let $h(x)=\lambda f(x) + \mu g(x)$.
Then of course $T(h(x))=h(x+1)=...$.

So does this show it: