Graph the following function using the technique of shifting, compressing, strecthing and/or reflecting

Find domain and range of the function
g(x)=2sqrtx-3+7
domain=___ (interval notation)
range=_____ (interval notation)

Start with √x

shift right by 3: √(x-3)
stretch by 2: 2√(x-3)
shift up 7: 2√(x-3)+7

adjust the domain and range for each transformation.

To graph the function g(x) = 2√x - 3 + 7 using shifting, compressing, stretching, and reflecting, we need to understand the transformations of the basic square root function, f(x) = √x.

Shifting: The function g(x) = 2√x - 3 + 7 shifts the graph of the basic square root function f(x) = √x three units to the left and seven units up.

Stretching/Compressing: The coefficient 2 in front of the square root function stretches the graph vertically by a factor of 2.

Reflecting: There is no reflection mentioned in this function, so no reflection will be applied.

Now that we know how to graph the function, let's determine the domain and range.

Domain refers to all the possible values of x for which the function is defined. In this case, since we have a square root function, the value inside the square root (√x) needs to be greater than or equal to zero to avoid taking the square root of a negative number, which is undefined. Therefore, we have √x ≥ 0.

Solving the inequality, we find x ≥ 0. So, the domain of the function g(x) = 2√x - 3 + 7 is [0, ∞) in interval notation.

Range refers to all the possible values of y (or g(x)) that the function can produce. Since the square root function (√x) always produces non-negative outputs, the range of g(x) = 2√x - 3 + 7 is [4, ∞) in interval notation, as the lowest value it can reach is 4 when x = 0.

To graph the function, start by plotting the basic square root function f(x) = √x. Then, apply the transformations:

1. Shift three units to the left: Each point (x, y) on the basic graph is shifted three units to the left to become (x + 3, y).
2. Stretch vertically by a factor of 2: Multiply each y-coordinate by 2.

By applying these transformations, you can plot the points and sketch the graph accordingly.