example 2 sketch f(x) { x² + 4 for x ≤ -1 -x² for x > -1 }

Explanation:

Step1: Analyze the first - part of the function

For (y = x^{2}+4) when (x\leq - 1). This is a parabola (y = x^{2}) shifted up by 4 units. When (x=-1), (y=(-1)^{2}+4=5). As (x\to-\infty), (y\to+\infty).

Step2: Analyze the second - part of the function

For (y=-x^{2}) when (x > - 1). This is an inverted parabola. When (x=-1), (y=-(-1)^{2}=-1) (but this value is not included for this part of the piece - wise function). As (x\to+\infty), (y\to-\infty).

Step3: Sketch the graph

Plot the points and draw the two parts of the function. For (y = x^{2}+4) ((x\leq - 1)), start from the point ((-1,5)) and draw the left - hand side of the parabola. For (y=-x^{2}) ((x > - 1)), draw the part of the inverted parabola to the right of (x=-1) (with an open - circle at (x = - 1) for this part).

Answer:

A graph with a parabola (y = x^{2}+4) for (x\leq - 1) and an inverted parabola (y=-x^{2}) for (x > - 1) is sketched as described above.