Why is a square root not a linear transformation?

I suppose you could start with the definition of a linear transformation.

I think you have it right that $T(cu)\ne c T(u)$ for scalar $c$.

You might look at the other property of a linear transformation:

Is $T(x+y) = T(x)+T(y)$?

(i.e. does $\sqrt{x+y}=\sqrt{x}+\sqrt{y}$?)

Edit: If we want to be more abstract. (I doubt this is what the question is aiming for, but others have pointed this out.) We can define addition in our vector space as multiplication and scalar multiplication as exponentiation. This would work for $\mathbb{R_{>0}$. We can then show that the $f$ does work. It is likely the problem you are working on intends the reals in which case you can ignore everything past "edit".

Others have gone the symbolic route... However, it is sufficient to show one (valid) assignment of the variables that fails to achieve equality. For instance, if $x = 1$, we would need to show that $\sqrt{1+1^2} = \sqrt{2}$ does not equal $\sqrt{1} + \sqrt{1^2} = 1 + 1 = 2$. And this should be evident.

As Matthew Gunn pointed out, the question definitely seems to be asking you to prove $\sqrt{1+x^2}≠1+x$ in order to disprove the statement: "$\sqrt{X+Y}=\sqrt{X}+\sqrt{Y}$ for each $(X,Y)$ in $R^2$".
You may start by supposing $\sqrt{X+Y}=\sqrt{X}+\sqrt{Y}$ for each $(X,Y)$, applying that to $X=1$ and $Y=x^2$, and quickly reaching an absurdity (for example, if $x=y$ then $x^2=y^2$).

You then end up in contradiction with the very definition of a linear transformation and can conclude on that.

Suppose that: $\sqrt{1+x^2}=1+x.$

Then: $1+x^2=1+x^2+2x.$

Then: $x=0$.

So, since in general $x \in \mathbb{R}$, the hypothesis cannot be true.

I'll go for something different.

A one variable linear transformation must have a vector space as it's domain, yet the square root has a restricted domain, only the non-negative real numbers. The non-negative real numbers (under addition) is not a vector space (can you see why?).

If linearity is understood over the field of reals, then since $f(-x)$ does not exist for $x> 0$, case closed. Even if we restrict to a weak form of linearity, $\sqrt{cx} = c\sqrt{x}$ for $x> 0$ only when $c=1$.

Using the given hint, $(\sqrt{1+x^2})^2 - (1+x)^2 = 2x$, hence both quantities on the LHS are equal only when $x=0$.

These are two reasons to conclude that $f$ is not linear. Each one is sufficient, since a counter-example suffices.

In passing, one sometimes linearize this function around $1$. A standard first order development exists when $y\sim 0$: $\sqrt{1+y} \sim 1+\frac{y}{2} $.

The statement

Statement 1: The square root is not a linear transformation.

is not generally true, at least as it stands.

According to the most often applied definition (see, e.g., Wikipedia), a linear transformation $f : V \to W$ is a mapping between two vector spaces/ modules $V$ and $W$ over the same field/ ring.

The difficulty is to acknowledge that the operations on $V$ and $W$ can well be different from usual addition and usual multiplication, even if the symbols $+$ and $\cdot$ are often used to denote vectos addition and and scalar multiplication of an abstract vector space. The terms of the two operation are themselves suggestive, but misleading. Indeed, when the vector space under consideration is the concrete vector space $\mathbb{R}$, then vector addition and scalar multiplication agree with usual addition and usual multiplication; but there are numerous examples in which this is not the case. An important and simple example is the vector space of the positive-real numbers, $\mathbb{R}_{>0}$, in which multiplication and exponentiation take on the role of vector addition and scalar multiplication.

Now consider the OP's square-root function $f$. The domain of $f$ cannot be $\mathbb{R}$, because the square root is not defined for negative values (let us leave aside the complex-valued case). Therefore, it is questionable if the operations of $\mathbb{R}$ are, afer all, validly used to proof that Statement 1 is true. It may well be argued that the proofs of Statement 1 are flawed, because the requirements of the definition of a liear map are not met. The appendage in the OP's question, $\sqrt{1 + x^2} \neq 1 + x$, is actually irrelevant for this matter.

In fact, one can easily prove, meeting all requirements of the definition of a linear map, the converse of Statement 1:

Proposition 1: The map $f : \mathbb{R}_{>0} \to \mathbb{R}_{>0}, \, x \mapsto \sqrt{x}$ is a linear transformation.

Proof: Let $x,y \in \mathbb{R}_{>0}$, and let $\lambda \in \mathbb{R}$. Then $$ \begin{array}{rclr} f(x \cdot y) =& \sqrt{x \cdot y} = \sqrt{x} \cdot \sqrt{y} &= f(x) \cdot f(y) & \text{(preservation of vector addition)} \\ f(x^\lambda) =& \sqrt{x^\lambda} = \left( \sqrt{x} \right)^\lambda &= f^\lambda(x) & \text{(preservation of scalar multiplication)} \end{array} $$ $$\tag*{$\square$}$$

A transformation $T$ is linear if and only if $T(ax+b)=aT(x)+b, \forall a,b$. Applied to $T(x)=\sqrt{x}$ this would mean that for any $ a,b$ it would hold that $\sqrt{ax+b}=a\sqrt{x}+b$. Let's proof that this can not hold using the technique ''proof by contradiction'', so assume that for $\forall a,b$ it holds that $\sqrt{ax+b}=a\sqrt{x}+b$, then it would hold that (square both sides) $ax + b = a^2 x + 2ab\sqrt{x} + b^2$, so only if $a^2=0$ and $ab=0$ and $b=b^2$ or only if $a=0$ and $b=b^2$ or only if $a=0$ and $b \in \{0,1\}$. This is a contradiction because we intially assumed that it holds $\forall a,b$ and we find that it does only hold for $a=0$ and $b \in \{0,1\}$.