Find The Range Of The Following Piecewise Function.

Findingthe range of a piecewise function requires careful analysis of each segment and then combining the results; this guide explains how to find the range of the following piecewise function step by step, offering a clear roadmap that works for any similar problem.

Introduction to Piecewise Functions

A piecewise function is defined by multiple sub‑functions, each applying to a specific interval of the independent variable. The overall function “pieces” together these pieces, and its range is the set of all possible output values it can produce. Because each piece may have its own behavior—linear, quadratic, constant, or even more complex—the range is not always obvious. Understanding the structure of the function and the behavior of each piece is essential before attempting to determine the final range.

What Makes a Function Piecewise?

• Multiple definitions: The function is described by two or more formulas.
• Domain partitions: Each formula covers a distinct interval, often expressed with inequality symbols.
• Continuity considerations: Some pieces may meet at boundary points, while others may leave gaps or jumps.

Steps to Find the Range of a Piecewise Function

Step 1: Identify Each Piece and Its Domain Begin by listing every formula that makes up the function together with the interval it governs. Write these out explicitly, for example:

• Piece 1: (f(x)=2x+1) for (x\le 0)
• Piece 2: (f(x)=x^{2}) for (0<x\le 3)
• Piece 3: (f(x)=\frac{1}{x}) for (x>3)

Mark the endpoints clearly, noting whether they are included (closed interval) or excluded (open interval). This step prevents accidental inclusion or exclusion of values later on.

Step 2: Determine the Range of Each Individual Piece

For each sub‑function, find its own range over the assigned domain. Common strategies include:

• Algebraic manipulation: Solve (y = \text{formula}) for (x) and examine constraints.
• Calculus tools (if allowed): Use derivatives to locate maxima, minima, or monotonic behavior.
• Graphical intuition: Sketch a quick plot to visualize the behavior at the boundaries.

Example: For (f(x)=2x+1) with (x\le 0), the expression is linear and decreasing as (x) becomes more negative. As (x) approaches (-\infty), (2x+1) also approaches (-\infty); at (x=0) (included), the value is (1). Hence the range of this piece is ((-\infty,1]).

Step 3: Combine the Individual Ranges

After obtaining the range for each piece, merge them while respecting any overlaps or gaps. Use set‑theoretic notation such as union ((\cup)) to combine the intervals. Pay special attention to:

• Closed vs. open endpoints: If a boundary value is attained by one piece, it belongs to the overall range even if another piece excludes it.
• Discontinuities: Gaps in the combined range indicate values that the function never takes.

Example continuation: The second piece (x^{2}) for (0<x\le 3) yields values from just above (0) up to (9). Since (x=0) is excluded, the lower bound is open; the upper bound is closed because (x=3) is included, giving ((0,9]). The third piece (\frac{1}{x}) for (x>3) produces values that decrease toward (0) as (x) grows, but never reaches (0). At (x=3^{+}) the value is slightly less than (\frac{1}{3}), and as (x\to\infty), the output approaches (0) from the positive side. Thus its range is ((0,\frac{1}{3})).

Finally, combine all three ranges:

[ (-\infty,1];\cup;(0,9];\cup;(0,\tfrac{1}{3}) = (-\infty,9] . ]

Notice that the overlapping positive intervals merge into a single continuous segment, while the negative infinity side remains unchanged.

Example: Finding the Range of a Sample Piecewise Function

Consider the following concrete piecewise function:

[ g(x)=\begin{cases} \displaystyle \sqrt{x} & \text{if } 0\le x\le 4,\[6pt] \displaystyle -\frac{1}{x-5} & \text{if } x>4. \end{cases} ]

Writing the Function Clearly

• First piece: (g(x)=\sqrt{x}) defined on the closed interval ([0,4]).
• Second piece: (g(x)=-\frac{1}{x-5}) defined for (x>4) (note the vertical asymptote at (x=5)).

Solving for Each Interval

1. First interval ([0,4]):

   • The square‑root function is increasing, so its minimum occurs at (x=0) giving (0), and its maximum at (x=4) giving (2).
   • Hence the range for this piece is ([0,2]).

2. Second interval ((4,\infty)) excluding (x=5):

   • Rewrite the expression as (-\frac{1}{x-5}).
   • As (x) approaches (4^{+}) from the right, (x-5) approaches (-1), making the whole fraction approach (1). Since (x=4) is not included, the value (1) is not attained.
   • As (x) moves just past (5), the denominator becomes a small positive number, causing the fraction to drop to large negative values. Thus the function shoots toward (-\infty) as (x\to5^{+}).

The process of analyzing ranges becomes more nuanced when dealing with complex piecewise definitions, especially when considering both asymptotic behavior and boundary conditions. Each segment must be carefully evaluated for its contribution to the overall output spectrum. For instance, the second piece’s transformation reveals a clear path from near positive infinity down to negative extremes, emphasizing the importance of tracking limits as boundaries are approached.

When revisiting the combined range, we observe that certain values are effectively excluded or included based on strict inclusion of endpoints. The careful merging of intervals highlights how discontinuities and limiting behaviors shape the final outcome. This exercise reinforces the value of set‑theoretic tools in organizing infinite or piecewise-defined outputs.

In summary, understanding these relationships clarifies not only the mathematical structure but also the underlying logic behind function behavior. This approach equips us to tackle similar problems with precision and confidence.

In conclusion, a systematic examination of boundaries, asymptotes, and interval intersections yields a comprehensive range that captures all attainable values. This method serves as a reliable framework for further exploration of complex mathematical models.

Analyzing the Combined Range

Combining the ranges from each interval, we see that the function’s output extends from 0 up to 2 within the interval [0, 4]. As x approaches 4 from the right, g(x) approaches 1, but never actually reaches it. Simultaneously, as x moves beyond 4, g(x) decreases without bound, heading towards negative infinity. The vertical asymptote at x = 5 creates a sharp cutoff, preventing the function from ever taking on values less than a very large negative number. Therefore, the overall range of g(x) is ((-\infty, 2]).

Investigating Key Points and Behavior

Let’s consider the behavior at the boundary point x = 4. Since g(x) is defined as √x for x ≤ 4, we have g(4) = √4 = 2. However, as x approaches 4 from the right, g(x) approaches 1. This creates a “hole” in the graph at x = 4, where the function is undefined. The vertical asymptote at x = 5 is crucial; it dictates that the function will become increasingly negative as x gets closer to 5 from the right.

Visualizing the Function

A quick sketch of the function would reveal this behavior clearly. The first piece is a simple increasing square root curve. The second piece is a hyperbola-like curve that rapidly decreases as x increases beyond 4. The “hole” at x = 4 and the asymptote at x = 5 are essential features to note.

Conclusion

The piecewise function g(x) presents a compelling example of how careful consideration of intervals, boundaries, and asymptotes is vital for accurately determining its range and understanding its overall behavior. The combination of the square root function and the reciprocal function creates a function with a defined interval, a point of discontinuity, and a vertical asymptote, resulting in a range that extends from negative infinity to 2. By systematically analyzing each component and their interactions, we gain a complete picture of this function’s characteristics, reinforcing the importance of a rigorous approach to function analysis in mathematics.