Suppose That The Function F Is Defined As Follows

Suppose that the function f is defined as follows:

[ f(x)=\begin{cases} x^2 & \text{if } x<0,\[4pt] 2x+1 & \text{if } 0\le x\le 3,\[4pt] \sqrt{x-3} & \text{if } x>3. \end{cases} ]

This opening statement is a typical way mathematicians introduce a piecewise‑defined function. In the sections that follow we will unpack what it means to suppose that the function f is defined as follows, explore its domain and range, examine continuity and differentiability, and see how such definitions appear in real‑world modeling.

Introduction

When a textbook or lecture begins with “Suppose that the function f is defined as follows,” it signals that the upcoming discussion will rely on a specific rule for assigning outputs to inputs. The phrase itself is not just a formality; it sets the stage for analyzing properties such as limits, derivatives, integrals, and graphical behavior. Understanding how to interpret and work with these definitions is a foundational skill in calculus, analysis, and applied mathematics. ---

Definition and Notation

A function f from a set A (the domain) to a set B (the codomain) assigns each element x in A exactly one element f(x) in B. When we write

[f(x)=\begin{cases} \text{expression}_1 & \text{if condition}_1,\ \text{expression}_2 & \text{if condition}_2,\ \vdots & \vdots\end{cases} ]

we are using piecewise notation. Each line gives a formula that is valid only on the subset of the domain where its condition holds. The conditions must be mutually exclusive and together cover the entire domain; otherwise the definition would be ambiguous or incomplete.

Key points to remember

• The domain is the union of all intervals (or sets) where the conditions are true.
• The range is the set of all output values produced by applying each piece to its respective sub‑domain.
• At the boundaries between pieces we must check the function’s value from the left and from the right to determine whether the function is well‑defined there.

Piecewise Definitions: Why They Matter

Piecewise definitions arise naturally in many contexts:

1. Physical systems that change behavior at a threshold (e.g., a tax bracket, a material’s stress‑strain curve).
2. Signal processing where a signal is clipped, rectified, or filtered differently in different frequency bands. 3. Optimization problems where the objective function has different forms depending on decision variables.

By supposing that the function f is defined as follows, we create a controlled environment to study how these transitions affect mathematical properties.

Exploring the Example Function

Let’s analyze the concrete example introduced earlier:

[ f(x)=\begin{cases} x^2 & \text{if } x<0,\[4pt] 2x+1 & \text{if } 0\le x\le 3,\[4pt] \sqrt{x-3} & \text{if } x>3. \end{cases} ]

Domain

• For (x<0) the expression (x^2) is defined for all real numbers, so this piece contributes ((-\infty,0)).
• For (0\le x\le 3) the linear expression (2x+1) is defined on the closed interval ([0,3]).
• For (x>3) the square‑root (\sqrt{x-3}) requires (x-3\ge0), i.e., (x\ge3); combined with the strict inequality (x>3) we get ((3,\infty)).

Thus the overall domain is ((-\infty,\infty)), the entire real line.

Range

• On ((-\infty,0)), (x^2) yields values in ((0,\infty)). As (x\to0^{-}), (x^2\to0^{+}); as (x\to -\infty), (x^2\to\infty).
• On ([0,3]), (2x+1) runs from (2\cdot0+1=1) to (2\cdot3+1=7). Hence this piece contributes ([1,7]).
• On ((3,\infty)), (\sqrt{x-3}) starts at (\sqrt{0^{+}}=0^{+}) and increases without bound, giving ((0,\infty)). Combining the three ranges we obtain ([0,\infty)). Notice that the value 0 is attained only as a limit from the left of the first piece and from the right of the third piece; the function itself never actually outputs 0 because the first piece is open at 0 and the third piece is open at 3. However, the limit as (x\to0^{-}) and as (x\to3^{+}) both approach 0, so the infimum of the range is 0.

Continuity

To test continuity we examine the points where the definition changes: (x=0) and (x=3).

• At (x=0): - Left‑hand limit: (\displaystyle \lim_{x\to0^{-}} f(x)=\lim_{x\to0^{-}} x^2 =0).

  • Right‑hand limit: (\displaystyle \lim_{x\to0^{+}} f(x)=\lim_{x\to0^{+}} (2x+1)=1).
  • Function value: (f(0)=2\cdot0+1=1).

  Since the left‑hand limit (0) does not equal the right‑hand limit (1), the function has a jump discontinuity at (x=0).

• At (x=3): - Left‑hand limit: (\displaystyle \lim_{x\to3^{-}} f(x)=\lim_{x\to3^{-}} (2x+1)=7).

  • Right‑hand limit: (\displaystyle \lim_{x\to3^{+}} f(x)=\lim_{x\to3^{+}} \sqrt{x-3}=0).