g(x)=2√x-3 + 7
One important thing to remember in these problems is that 
we must shift vertically UP or DOWN last.

We start with

y = √x


Its domain is [0,)
Its range is [0,)

Then we stretch it vertically by a factor of 2 by multiplying
the right side by 2 and we get

y = 2√x


Its domain is [0,)
Its range is [0,)
Note that these did not change on this step


Then we shift it horizontally to the right by 3 units by 
replacing x by (x-3) in the right side and we get:

y = 2√x-3


Its domain is now [3,)
Its range is [0,)
Note that the domain shifted 3 to the right but
the range did not change on this step.

Then finally we shift it vertically up by 7 units by adding 7 
to the right side.:

y = 2√x-3 + 7


And that is the graph of

g(x) = 2√x-3 + 7

Its domain is [3,)
Its range is [7,)
Note that the domain did not change from the previous step,
but the range shifted 7 units upward.

Edwin


 

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