Confusing notation related to functions

Question

The notation given is:

$$g(x)= \min\lbrace f(t): -3\leq t \leq x, -3\leq x \leq0\rbrace$$

What does this even mean? Is this just another way of saying that $t\in[0,3]$ ?

Answer 1

It means that $g(x) $ is the smallest value of $f(t)$ where $t \in [-3, x] $ and that $x \in [-3, 0]$. In essence, the value that $t$ can take is dependent on the value of $x$ as well.

Maybe, you could also shed light on where you encountered the above function.

Answer 2

That seems like a mistake. Probably what is meant is $$g(x)= \min\lbrace f(t): -3\leq t \leq x\rbrace, -3\leq x \leq0$$

That means, for each $x$ you look at the minimum value of $f$ in the region $-3\leq t \leq x$.

The domain for $g$ is $[-3,0]$.