Answer:
domain is
[
3
,
∞
)
and our range is
(
−
∞
,
1
]
Step-by-step explanation:
Let's look at the parent function:
√
x
The domain of
√
x
is from
0
to
∞
. It starts at zero because we cannot take a square root of a negative number and be able to graph it.
√
−
x
gives us
i
√
x
, which is an imaginary number.
The range of
√
x
is from
0
to
∞
This is the graph of
√
x
graph{y=sqrt(x)}
So, what is the difference between
√
x
and
−
2
⋅
√
x
−
3
+
1
?
Well, let's start with
√
x
−
3
. The
−
3
is a horizontal shift, but it is to the right, not the left. So now our domain, instead of from
[
0
,
∞
)
, is
[
3
,
∞
)
.
graph{y=sqrt(x-3)}
Let's look at the rest of the equation. What does the
+
1
do? Well, it shifts our equation up one unit. That doesn't change our domain, which is in the horizontal direction, but it does change our range. Instead of
[
0
,
∞
)
, our range is now
[
1
,
∞
)
graph{y=sqrt(x-3)+1}
Now let's see about that
−
2
. This is actually two components,
−
1
and
2
. Let's deal with the
2
first. Whenever there is a positive value in front of the equation, it is a vertical stretching factor.
That means, instead of having the point
(
4
,
2
)
, where
√
4
equals 2
, now we have
√
2
⋅4 equals 2
. So, it changes how our graph looks, but not the domain or the range.
graph{y=2 * sqrt(x-3)+1}
Now we've got that −
1
to deal with. A negative in the front of the equation means a refection across the
x
-axis. That won't change our domain, but our range goes from
[
1
,
∞
) to
(
−
∞
,
1
]
graph{y=-2sqrt(x-3)+1}
So, our final domain is
[
3
,
∞
)
and our range is
(
−
∞
,
1
]
