Heuristics as to when $\frac{x-c_1}{c_2-c_1}$ and similar forms are linear and form a line?

Question

Heuristics as to when $f(x)=\frac{x-c_1}{c_2-c_1}$ and similar forms are linear and form a line?

By similar forms I mean that I started to speculate as to how much one can "add stuff" to the above simple case and still know that the function is a linear function and its graph is a line.

Intuitively one can add, subtract and do multiplications by constants. One could also add linear functions to it and still retain linearity. But linearity would surely be broken if one adds a nonlinear function $\eta(x)$ like $h(x)=f(x)+\eta(x)$?

So additivity and homogeneity do not suffice, but they have some constraints as to what can be added and what can be used as multiplier. What are these constraints?

Answer 1

The affine functions of one variable are of the form

$$ax+b$$ where $a$ and $b$ are constants.

Nothing more to say.

Answer 2

Generalization

Consider the function $f:\Bbb R^n\to \Bbb R$ as follows $$f(x)={a^Tx+b\over c^Tx+d}$$where $a,c\in \Bbb R^n$ and $b,d\in\Bbb R$. The case where $a=0$ , $b=0$ is trivial, so we investigate the rest. If $f(x)$ is affine, then $e\in \Bbb R^n$ , $f\in \Bbb R$ exist such that $$f(x)=e^Tx+f$$by substitution we obtain $${a^Tx+b\over c^Tx+d}=e^Tx+f\\a^Tx+b=x^Tce^Tx+df+(fc+de)^Tx$$which yields to $$b=df\\ce^T=0\\fc+de=a$$some cases are worth studying:

---

Case 1 $d=0$

In this case $b=0$ and we obtain $a=fc$ with $f\ne 0$ and we have $$f(x)={a^Tx\over c^Tx}=f{c^Tx\over c^Tx}$$ which is not affine over $\Bbb R^n$ since the function is not defined for $c^Tx=0$. Therefore the case $d=0$ doesn't lead to affinity.

---

Case 2 $d\ne 0$ , $c=0$

We obtain$$f={b\over d}\\de=a$$which means that affinity happens in this case without extra assumptions.

---

Case 3 $d\ne0$ and $c\ne 0$

The equations lead to $$f={b\over d}\\ec^T=0\\fc+de=a$$from $$ec^T=0$$ we obtain $ec^Tc=e\cdot ||c||^2=0$ and since $c\ne 0$ we have $e=0$ therefore $$f={b\over d}\\fc=a$$ and since $c^Tx+d=0$ has infinitely many roots, then this case either doesn't lead to affinity.

Conclusion

The only case leading to affinity is $c=0$ and $d\ne 0$.