Q.The relation $f$ is defined by

[!FORMULA] $f(x)=\begin{cases} x^2, & 0 \le x \le 3 \\ 3x, & 3 \le x \le 10 \end{cases}$

The relation $g$ is defined by

[!FORMULA] $g(x)=\begin{cases} x^2, & 0 \le x \le 2 \\ 3x, & 2 \le x \le 10 \end{cases}$

Show that $f$ is a function and $g$ is not a function.

What is a Piecewise Function? — The Intuition

Imagine you're describing how much a taxi ride costs. The fare might be: ₹25 for the first kilometer, then ₹15 for every kilometer after that. That's not a single, simple rule — the rule changes depending on how far you've gone. That's exactly what a piecewise function captures: a function whose rule is made of different "pieces," each applying to a different part of the input.

In everyday life, piecewise rules are everywhere:

• Income tax slabs (different rates for different income ranges)
• Mobile data plans (different speeds after a limit)
• Postage rates (different costs for different weights)

A piecewise function lets you write all these different rules in one clean mathematical statement.

The Precise Definition

A piecewise function is a function defined by multiple sub-functions, each applying to a specific interval (or "piece") of the domain.

Here's the standard notation:

Where:

• $f_1, f_2, \dots, f_n$ are the sub-functions (each is a rule)
• $D_1, D_2, \dots, D_n$ are disjoint intervals that together cover the entire domain
• Each input $x$ belongs to exactly one of these intervals

A Concrete Example

Let's write the taxi fare example properly. Suppose the first kilometer costs ₹25, and every subsequent kilometer costs ₹15 per km. For a ride of $x$ kilometers:

Let's test it:

• For $x = 0.5$ km: $f(0.5) = 25$ (first piece)
• For $x = 1$ km: $f(1) = 25$ (first piece, includes the endpoint)
• For $x = 3$ km: $f(3) = 25 + 15(3 - 1) = 25 + 30 = 55$ (second piece)

Notice how the second piece uses $x - 1$ — that's because the ₹15 rate only applies to the distance beyond the first kilometer.

Common Pitfalls (Watch Out!)

Why This Matters …