Name the property $f(x) \ge x$ It's a really one of the simplest properties you could imagine for a function. But I haven't been able to find a name for it. What do you call a function $f$ with the following property:

$$f(x) \ge x$$

Note, this is not monotonicity. Monotonicity is the following property:

$$x \ge y \implies f(x) \ge f(y)$$
As mentioned, this property is called expansion, as the element $x$ can be expanded, or stretched-out and thus contained in, the element $f(x)$. 

However, this property is also termed reflexivity.

Why? Well, recall that a relation is reflexive if it relates identical elements.
More precicely,

$\;\;\; R \text{ reflexive}
\\\equiv \forall x:: \; x \, R \, x
\\\equiv \forall x,y:: \; x = y \implies x \, R \, y
\\\equiv \forall x,y:: \; x \, I \, y \implies x \, R \, y \text{, where $I$ is the identity relation: $I=\{(x,y)\,|\,x=y\}$}
\\\equiv I \subseteq R
$

Even more so, recall that an ordering $\leq$ on elements can be lifted to functions by $$f \overset{.}{\leq} g \,:\equiv\, (\forall x :: f \ x \leq g \ x)$$
With this, and letting $1$ be the identity function $x \mapsto x$, we have
$$f \text{ reflexive } \ \equiv \ 1 \overset{.}{\leq} f$$

Hope that helps!
I call it the opposite of a contraction; ie the concave variant; it would follow from star concavity at $x_0=0$ where $F(x_0=0)\geq 0$with
now as $$F(1)=1 \land \text{dom(F)}=[0,1]$$.

star concavity at 0 $$\forall (t\in [0,1]):\forall (x \in [0,1]):F((1-t)\times x_0+t\times y )\geq t\times F(y)+(1-t)\times F(x_0)$$.

ie
$$\to F(t\times y)=F( (1-t)\times 0+ t\times y)\geq (1-t)\times F(0)+t\times f(y)$$

As $(t,1-t)\in [0,1]\land F(x_0=0)\geq 0$

$$\to F(t\times y)\geq (1-t)\times F(0)+t\times f(y)\geq  t \times F(y)$$ 

star concavity  at $0$ with $F(0)\geq 0(B)$
$$\to (B)F(t\times y)\geq t\times F(y)$$ 

(2) now as $$F(1)=1 \land \text{dom(F)}=[0,1]$$.

$$t=x \to F(x)=F(t)$$
$$F(x)=F(t)= F(1 \times t)\to F(x)=F(1\times t)$$
concavity

$$\to F(x)=F(1\times t)\geq t\times F(1)$$

substitution of $F(1)=1$ 
$$\to F(x)\geq t\times F(1)=1\times t=t$$
$$\to F(x)\geq t $$=

where t=x so:
$$\to F(x)\geq x$$