Functions and transformations

Question

A function, $y = f(x)$, originally has a domain of $x ≥ 4$ and a range of $y ≤ 1$.

Determine the new domain and range of $y=-2f(-x+5)+1$ after applying all transformations.

(Try sketching the graph and applying the transformations,)

I am stuck with even starting this question without having the original function. But based on the domain and range of the original I think this is a square root parent function. I am at loss; please help. This question has been bugging me and I am sure I am overthinking.

Answer 1

In keeping with your idea about using a square root parent function, I put into the graphing site Desmos a function $f$ with the desired domain and range: link.

You will see there not only the original graph, but also a sequence of transformations that get to the final function that you are supposed to be considering. These are color coded, and you can click the color next to each function input to hide it.

Can you see how each of them affects the parent graph?

Here is an image for completeness:

Answer 2

From the domain and the range of $f(x)$, we may know that $$-x+5 \geq 4, ~~~~~~ f(-x+5) \leq 1.$$

Thus, for the new function, the domain is $$x \leq 1,$$ and the range is $$y=-2f(-x+5)+1 \leq -2 \cdot 1+1=-1.$$