Advanced functions

When one function is built from another by shifts, like $g(x) = f(x - 2) + 2$, the graph of $g$ is just the graph of $f$ moved.
A change inside the function shifts it horizontally, and a change outside shifts it vertically.

Read the two shifts separately.
In $g(x) = f(x - 2) + 2$, the $(x - 2)$ moves the graph right by $2$, and the $+2$ moves it up by $2$.
So if the graph of $f$ has its vertex at $(1, 0)$, the graph of $g$ has its vertex at $(1 + 2,\ 0 + 2) = (3, 2)$; every point of $f$ moves the same way.

To go the other way and recover $f$ from $g$, undo the shifts.
Since $g$ is $f$ moved right $2$ and up $2$, the graph of $f$ is the graph of $g$ moved left $2$ and down $2$.
So a vertex of $g$ at $(3, 2)$ came from a vertex of $f$ at $(1, 0)$.

You can also read a single value of $f$ straight from the graph of $g$.
Because $g(x) = f(x - 2) + 2$, putting $x = 3$ gives $g(3) = f(1) + 2$, so $f(1) = g(3) - 2$.
Read $g(3)$ off the graph and subtract $2$: match the input, then adjust for the outside shift.

The graph of $g$ is shown, where $g(x) = f(x + 3) - 1$.

What is the vertex of the graph of $f$?
The $(x + 3)$ moves $f$ left $3$ and the $-1$ moves it down $1$, so $f$ is recovered by moving $g$ right $3$ and up $1$.
The vertex of $g$ is $(-2, 2)$, so the vertex of $f$ is $(-2 + 3,\ 2 + 1) = (1, 3)$.

The graph of $g$ is shown, where $g(x) = f(x + 1) - 3$.

The graph of $f$ is a parabola; what is its vertex?
The $(x + 1)$ moves $f$ left $1$ and the $-3$ moves it down $3$, so $f$ is $g$ moved right $1$ and up $3$.
The vertex of $g$ is $(2, -2)$, so the vertex of $f$ is $(2 + 1,\ -2 + 3) = (3, 1)$.

The graph of $g$ is shown, where $g(x) = f(x - 3) + 1$.

What is the value of $f(1)$?
Since $g(x) = f(x - 3) + 1$, set $x - 3 = 1$, so $x = 4$; then $g(4) = f(1) + 1$, which gives $f(1) = g(4) - 1$.
From the graph $g(4) = 4$, so $f(1) = 4 - 1 = 3$.