A function $f(x, y)$ is said to be homogeneous of degree $k$ if $f(tx, ty) = t^k f(x, y)$ for some real number $k$ and all $t > 0$.

To identify a function that is not homogeneous, we look for functions where this scaling property does not hold. Common characteristics of non-homogeneous functions include:
1. **Presence of a constant term:** A term that does not involve $x$ or $y$.
2. **Terms with different total degrees:** For polynomial functions, if the sum of the powers of $x$ and $y$ in each term is not the same.
3. **Transcendental functions (e.g., exponential, logarithmic) that do not scale appropriately:** For instance, $e^x$, $\log(x+y)$, or $\sin(x)$ are generally not homogeneous unless their arguments are homogeneous of degree 0.

Let's consider a common example of a function that is not homogeneous:
The function $f(x, y) = x^2 + y + 1$ is not a homogeneous function.

**Step-by-step verification:**
1. **Recall the definition of a homogeneous function:** $f(tx, ty) = t^k f(x, y)$.
2. **Substitute $tx$ for $x$ and $ty$ for $y$ into the function $f(x, y) = x^2 + y + 1$:**
$$f(tx, ty) = (tx)^2 + (ty) + 1$$
3. **Simplify the expression:**
$$f(tx, ty) = t^2x^2 + ty + 1$$
4. **Compare $f(tx, ty)$ with $t^k f(x, y)$:**
If $f(x, y)$ were homogeneous, we should be able to factor out $t^k$ such that the remaining expression is $x^2 + y + 1$.
However, we cannot write $t^2x^2 + ty + 1$ as $t^k(x^2 + y + 1)$ for any constant $k$. For instance, the term $1$ does not scale with $t$, and the term $ty$ scales differently from $t^2x^2$.

Since $f(tx, ty) \ne t^k f(x, y)$ for any real number $k$, the function $f(x, y) = x^2 + y + 1$ is not a homogeneous function.