need help for solving for a radical function

The parent function of [tex]f(x)=2\sqrt{x+4}[/tex] is [tex]f(x)=\sqrt{x}[/tex]. The 2 on the outside of the square root tells us that the parent graph will undergo a vertical stretch of 2. The +4 is really -(-4) so it is a negative four, therefore the parent function will horizontally shift to the left by 4 units.

Start by creating a t-chart with a few points from the parent function, [tex]f(x)=\sqrt{x}[/tex], on the left side then on the right side you'll transform the points from the parent function.

The left side of the t-chart should have the following points:

(0, 0)
(1, 1)
(4, 2)

Notice how there are no negative x-values because this is a square root function, and you cannot take the root of a negative number (it would be imaginary and not graphable in this kind of graph).

The right side of the t-chart should have the following points:

(-4, 0)
(-3, 2)
(0, 4)

I got these points by multiplying the y-values by the vertical stretch, 2, and subtracting 4 from the x-values to accomplish the horizontal stretch.

Note that everything outside of the "parentheses" (square root incl.) would be a vertical stretch/compression/shift, and everything inside the "parentheses" would be a horizontal stretch/compression/shift.

To solve your problem you would graph the right side of the t-chart because that is the transformed parent function.

Now for the domain and range:

The domain of the parent square root function is always going to be [tex]x\geq 0[/tex]. If you think about it, it makes sense since you cannot use any x-values less than 0 otherwise you'd come up with an imaginary number.

However, since this square root function is horizontally shifted 4 units to the left, the domain would start at -4. So the domain for this graph is: [tex]x\geq -4[/tex].

The range is the same; when you take the square root of a positive number you will not get a negative number. Therefore the range is also [tex]y\geq 0[/tex].

I've attached a picture of what the graph should look like.