How many functions $f:\mathbb{R}\to R\subseteq\mathbb{R}$ are there such that there is an $S\subseteq\mathbb{R}$ such that for any $\lambda\in S$ there exist $\alpha_\lambda,\beta_\lambda\in\mathbb{R}$ with $\lambda f(x)=f(\alpha_\lambda x+\beta_\lambda)$ for all $x\in\mathbb{R}$?

Obviously, $f(x)=x$ satisfies this with $R=S=\mathbb{R}$, $\alpha_\lambda=\lambda$, $\beta_\lambda=0$, and more generally $f(x)=|x|^s$ satisfies it with $R=S=(0;\infty)$, $\alpha_\lambda=\lambda^{1/s}$, $\beta_\lambda=0$.

Similarly, $f(x)=\exp(x)$ satisfies this with $R=S=(0;\infty)$, $\alpha_\lambda=1$, $\beta_\lambda=\log\lambda$, and more generally $f(x)=\exp(rx)$ satisfies it with $R=S=(0;\infty)$, $\alpha_\lambda=1$, $\beta_\lambda=\frac{1}{r}\log\lambda$.

But are there any other families of such functions? I see no obvious way to find them.

How many functions $f:\mathbb{R}\to R\subseteq\mathbb{R}$ are there such that there is an $S\subseteq\mathbb{R}$ containing an interval containing $1$ such that for any $\lambda\in S$ there exist $\alpha_\lambda,\beta_\lambda\in\mathbb{R}$ with $\lambda f(x)=f(\alpha_\lambda x+\beta_\lambda)$ for all $x\in\mathbb{R}$?

Obviously, $f(x)=x$ satisfies this with $R=S=\mathbb{R}$, $\alpha_\lambda=\lambda$, $\beta_\lambda=0$, and more generally $f(x)=|x|^s$ satisfies it with $R=S=(0;\infty)$, $\alpha_\lambda=\lambda^{1/s}$, $\beta_\lambda=0$.

Similarly, $f(x)=\exp(x)$ satisfies this with $R=S=(0;\infty)$, $\alpha_\lambda=1$, $\beta_\lambda=\log\lambda$, and more generally $f(x)=\exp(rx)$ satisfies it with $R=S=(0;\infty)$, $\alpha_\lambda=1$, $\beta_\lambda=\frac{1}{r}\log\lambda$.

But are there any other families of such functions? I see no obvious way to find them.

How many functions $f:\mathbb{R}\to R\subseteq\mathbb{R}$ are there such that there is an $S\subseteq\mathbb{R}$ such that for any $\lambda\in S$ there exist $\alpha_\lambda,\beta_\lambda\in\mathbb{R}$ with $\lambda f(x)=f(\alpha_\lambda x+\beta_\lambda)$ for all $x\in\mathbb{R}$?

Obviously, $f(x)=x$ satisfies this with $R=S=\mathbb{R}$, $\alpha_\lambda=\lambda$, $\beta_\lambda=0$, and more generally $f(x)=x^s$ satisfies it with $R=S=(0;\infty)$, $\alpha_\lambda=\lambda^{1/s}$, $\beta_\lambda=0$.

Similarly, $f(x)=\exp(x)$ satisfies this with $R=S=(0;\infty)$, $\alpha_\lambda=1$, $\beta_\lambda=\log\lambda$, and more generally $f(x)=\exp(rx)$ satisfies it with $R=S=(0;\infty)$, $\alpha_\lambda=1$, $\beta_\lambda=\frac{1}{r}\log\lambda$.

But are there any other families of such functions? I see no obvious way to find them.

How many functions $f:\mathbb{R}\to R\subseteq\mathbb{R}$ are there such that there is an $S\subseteq\mathbb{R}$ such that for any $\lambda\in S$ there exist $\alpha_\lambda,\beta_\lambda\in\mathbb{R}$ with $\lambda f(x)=f(\alpha_\lambda x+\beta_\lambda)$ for all $x\in\mathbb{R}$?

Obviously, $f(x)=x$ satisfies this with $R=S=\mathbb{R}$, $\alpha_\lambda=\lambda$, $\beta_\lambda=0$, and more generally $f(x)=|x|^s$ satisfies it with $R=S=(0;\infty)$, $\alpha_\lambda=\lambda^{1/s}$, $\beta_\lambda=0$.

Similarly, $f(x)=\exp(x)$ satisfies this with $R=S=(0;\infty)$, $\alpha_\lambda=1$, $\beta_\lambda=\log\lambda$, and more generally $f(x)=\exp(rx)$ satisfies it with $R=S=(0;\infty)$, $\alpha_\lambda=1$, $\beta_\lambda=\frac{1}{r}\log\lambda$.

But are there any other families of such functions? I see no obvious way to find them.