which of the following is the graph of this square root function? f(x) = √(x - 1) - 4

Answer:

1. First, find the domain of the function (y = \sqrt{x - 1}-4).
   • For the square - root function (\sqrt{u}), the expression inside the square root (u=x - 1) must be non - negative, i.e., (x-1\geq0). Solving for (x), we get (x\geq1).
2. Then, find the (y) - intercept (if it exists).
   • To find the (y) - intercept, set (x = 0). But (x = 0) is not in the domain of the function (y=\sqrt{x - 1}-4) since when (x = 0), (x-1=-1) and (\sqrt{-1}) is not a real number.
3. Next, find the (x) - intercept.
   • Set (y = 0), so (0=\sqrt{x - 1}-4).
   • Add 4 to both sides: (4=\sqrt{x - 1}).
   • Square both sides: (16=x - 1).
   • Solve for (x): (x=17).
4. Analyze the transformation of the parent function (y = \sqrt{x}).
   • The parent function (y=\sqrt{x}) has been shifted 1 unit to the right (because of (x-1) inside the square root) and 4 units down (because of (-4) outside the square root).
   • The starting point of the parent function (y = \sqrt{x}) is ((0,0)). After the transformation, the starting point of (y=\sqrt{x - 1}-4) is ((1,-4)).
   • Among the given graphs, the graph that starts at the point ((1,-4)) and has a domain of (x\geq1) is the correct one.

The graph that starts at the point ((1, - 4)) and has a domain (x\geq1) (the third graph if we assume the graphs are numbered from left - to - right as 1, 2, 3) is the graph of the function (f(x)=\sqrt{x - 1}-4).

Explanation:

Step1: Determine the domain

The expression inside the square root (x - 1\geq0), so (x\geq1).

Step2: Check for (y) - intercept

(x = 0) is not in the domain, so no (y) - intercept.

Step3: Find the (x) - intercept

Set (y = 0), solve (4=\sqrt{x - 1}), then (x = 17).

Step4: Analyze transformations

Shifted 1 unit right and 4 units down from (y=\sqrt{x}), starting point is ((1,-4)).