### Instruction: Given that $(-2,3)$ is on the graph of $y=f(x)$, find a point on the graph of $y=f(2x+1)+3$. Express your answer as an ordered pair $(a,b)$ where $a$ and $b$ are real numbers.### Response: I want to find a point on the graph of the transformed function, so I need to figure out how the transformation affects the $x$ and $y$ coordinates of any point on the original function.
The transformation has two parts: the input of the function is multiplied by $2$ and then increased by $1$, and the output of the function is increased by $3$.
This means that for any point $(x,y)$ on the original function, the corresponding point on the transformed function is $(\frac{x-1}{2}, y+3)$.
To see why, I can think of the inverse transformation: to get the input of the original function from the input of the transformed function, I need to subtract $1$ and then divide by $2$.
To get the output of the original function from the output of the transformed function, I need to subtract $3$.
Now I can apply this rule to the given point $(-2,3)$ on the original function.
The corresponding point on the transformed function is $(\frac{-2-1}{2}, 3+3) = (-\frac{3}{2}, 6)$.
Therefore, a point on the graph of $y=f(2x+1)+3$ is $(-\frac{3}{2}, 6)$.