explain how to graph the given piecewise - defined function. be sure to specify the type of endpoint each…

explain how to graph the given piecewise - defined function. be sure to specify the type of endpoint each piece of the function will have and why.\nf(x)=\begin{cases}-x + 3, &xlt2\\3, &2leq xlt4\\4 - 2x, &xgeq4end{cases}

explain how to graph the given piecewise - defined function. be sure to specify the type of endpoint each piece of the function will have and why.\nf(x)=\begin{cases}-x + 3, &xlt2\\3, &2leq xlt4\\4 - 2x, &xgeq4end{cases}

Answer

Answer:

  1. For (y=-x + 3) when (x<2):
    • First, find the (y) - intercept by setting (x = 0). Then (y=-0 + 3=3).
    • Find the (x) - intercept by setting (y = 0), so (0=-x + 3), which gives (x = 3).
    • Since the inequality is (x<2), when (x = 2), (y=-2 + 3=1). The endpoint at (x = 2) is an open - circle because the function is not defined for (x = 2) in this piece ((x<2)).
  2. For (y = 3) when (2\leq x<4):
    • This is a horizontal line at (y = 3). When (x = 2), the function value is (y = 3), and the endpoint at (x = 2) is a closed - circle because (x = 2) is included in the domain ((2\leq x<4)). When (x = 4), the function value is (y = 3), and the endpoint at (x = 4) is an open - circle because (x = 4) is not included in the domain ((2\leq x<4)).
  3. For (y=4 - 2x) when (x\geq4):
    • When (x = 4), (y=4-2\times4=4 - 8=-4). The endpoint at (x = 4) is a closed - circle because (x = 4) is included in the domain ((x\geq4)). As (x) increases, the function is a line with slope (-2). For example, if (x = 5), (y=4-2\times5=4 - 10=-6).

Explanation:

Step1: Analyze (y=-x + 3,x<2)

Find intercepts and endpoint.

Step2: Analyze (y = 3,2\leq x<4)

Determine horizontal line and endpoints.

Step3: Analyze (y=4 - 2x,x\geq4)

Find starting point and slope.