Questions: Graph the piecewise function given below. f(x)= (x+2)^2 for x<-1 x+3 for -1 ≤ x ≤ 3

Transcript text: Graph the piecewise function given below. \[ f(x)=\left\{\begin{array}{ll} (x+2)^{2} & \text { for } x<-1 \\ x+3 & \text { for }-1 \leq x \leq 3 \end{array}\right. \]

Solution

Solution Steps

Step 1: Graph the first piece

The first piece is defined as _f(x) = (x+2)²_ for _x < -1_. This is a parabola shifted two units to the left. Since the domain is restricted to x < -1, we only graph the portion of the parabola to the left of the vertical line x = -1. Plot the points (-2, 0), (-3, 1) and (-4, 4). Note that since x is strictly less than -1, there should be an open circle at the point (-1, 1).

Step 2: Graph the second piece

The second piece is defined as _f(x) = x + 3_ for _-1 ≤ x ≤ 3_. This is a line with a slope of 1 and a y-intercept of 3. Since the domain is restricted to -1 ≤ x ≤ 3, we only graph the portion of the line segment between x = -1 and x = 3 inclusive. Plot the points (-1, 2) and (3, 6). Note that since the interval includes -1 and 3, there should be closed circles at points (-1, 2) and (3, 6).

Final Answer The graph of the piecewise function

The final graph consists of the portion of the parabola _f(x) = (x+2)²_ to the left of x = -1 (with an open circle at x=-1) and the line segment _f(x) = x + 3_ from x = -1 to x = 3 (with closed circles at both endpoints).

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