A function $f:X→\mathbb{R}$ is said to be positive homogeneous of degree $k\in\mathbb{R}$ if $f(tx)=t^kf(x)$ for every $x\in X$ and every $t\in\mathbb{R}_{++}$.
For $X=\mathbb{R}_+$, the sample example is: $f(x)=M.x^k$ where $M\in\mathbb{R}$
Can you give me another example?
On
There are several classical economic functions that are known to be homogeneous of degree 1. First, we have the class of CES production functions, where $$f(x_1,\dots,x_n) = C\cdot \left(\sum_{i = 1}^n \left(\frac{x_i}{a_i}\right)^r\right)^{\frac1r}.$$ You can verify that this function is 1-homogeneous. Other famous functions are $f(x_1,\dots,x_n) = \min_i x_i$, and $f(x_1,\dots,x_n) = \max_i x_i$.
Other more interesting functions would be network flow functions. Given a directed graph $\Gamma = \langle V,E\rangle$, define a function $\mathit{flow}: \mathbb{R}_+^E \to \mathbb R_+$, as $\mathit{flow}(w_e)_{e \in E}$ being the maximum flow over $\Gamma$ when the edge weights are $(w_e)_{e \in E}$. There are many more, but I hope that this gives you a general idea of where to find them.
If $X = \mathbb R^3$, then $f(x,y,z)= |x^3 + x^2z + xyz-y^3|$ is an example with $k=3$.