Short Answer 
The analysis involves three types of functions: a quadratic function f(x) = (x + 2)¬≤ with a vertex at (-2, 0) for x < -1, an absolute value function f(x) = |x| + 1 with a vertex at (0, 1) for -1 ‚a§ x ‚a§ 1, and a square root function f(x) = -‚aox starting at (1, -1) for x > 1. Each function exhibits unique characteristics, such as parabolic, V-shape, and declining curves, respectively.
Step 1: Analyze the Quadratic Function
Begin by examining the quadratic graph of the function f(x) = (x + 2)² over the interval x < -1. This graph represents the parent function f(x) = x² shifted 2 units to the left. The characteristics of this graph include:
- A vertex at the point (-2, 0).
- It opens upwards and displays a classic parabolic shape.
Step 2: Investigate the Absolute Value Function
Next, look at the absolute value function defined as f(x) = |x| + 1 for the interval -1 ‚a§ x ‚a§ 1. This graph can be seen as a transformation of the parent function f(x) = |x|, which has been shifted upwards by 1 unit. Key features include:
- A vertex at the point (0, 1).
- The graph forms a V-shape, indicating an increase in both directions from the vertex.
Step 3: Examine the Square Root Function
Finally, for the interval x > 1, explore the graph of the square root function given by f(x) = -‚Äöaox. This represents a reflection of the parent function y = ‚Äöaox over the x-axis. Important aspects include:
- The graph starts at the point (1, -1) and decreases as x increases.
- It creates a decreasing curve bound to the x-axis, showcasing a decline to the right.