5. graph of the piece - wise function.\nf(x) = { |x - 3|+1, x≥0; 2√(x + 4), - 4≤x<0 }

5. graph of the piece - wise function.\nf(x) = { |x - 3|+1, x≥0; 2√(x + 4), - 4≤x<0 }

5. graph of the piece - wise function.\nf(x) = { |x - 3|+1, x≥0; 2√(x + 4), - 4≤x<0 }

Answer

Explanation:

Step1: Analyze the first - part for (x\geq0)

The function is (y = |x - 3|+1). When (x\geq3), (y=(x - 3)+1=x - 2). When (0\leq x<3), (y=-(x - 3)+1=-x + 4).

Step2: Analyze the second - part for (-4\leq x<0)

The function is (y = 2\sqrt{x + 4}). The domain of this part is ([-4,0)). When (x=-4), (y = 0); when (x = 0), (y=2\sqrt{4}=4).

Step3: Plot the functions

For (y=-x + 4) with (0\leq x<3), the (y) - intercept is (4) and the slope is (- 1). For (y=x - 2) with (x\geq3), the (y) - intercept is (-2) and the slope is (1). For (y = 2\sqrt{x + 4}) with (-4\leq x<0), start at the point ((-4,0)) and end at ((0,4)) with a curve shape typical of a square - root function.

To graph:

  1. For (y=-x + 4) ((0\leq x<3)): Plot the points ((0,4)) and ((3,1)) and draw a line segment between them.
  2. For (y=x - 2) ((x\geq3)): Plot the point ((3,1)) and then use the slope (m = 1) to draw a ray starting from ((3,1)) going to the right - hand side.
  3. For (y = 2\sqrt{x+4}) ((-4\leq x<0)): Plot the points ((-4,0)) and ((0,4)) and draw a smooth curve between them.

There is no single - value answer for graphing a piece - wise function. The steps above describe how to graph it.