Concept understanding — Piecewise Function Definition
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:
f(x)=⎩⎨⎧f1(x),f2(x),⋮fn(x),x∈D1x∈D2⋮x∈Dn
Where:
f1,f2,…,fn are the sub-functions (each is a rule)
D1,D2,…,Dn are disjoint intervals that together cover the entire domain
Each input x belongs to exactly one of these intervals
Important
A piecewise function is still one function — not several functions glued together. For every x in the domain, there is exactly one output f(x).
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:
f(x)={25,25+15(x−1),0<x≤1x>1
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!)
Watch out
Don't forget the domain conditions. A piecewise definition is incomplete without specifying which x values go with which rule. Writing just f(x)={x2,2x+1} is meaningless — you must say when each applies.
Watch out
Check the boundaries carefully. At the point where two pieces meet (like x=1 in the taxi example), the function must give only one output. If both pieces try to claim the same x, you have a problem — it's no longer a function.
f(x)=∣x∣ and f(x)=[x] are two standard functions with well-known graph shapes (a V and a staircase respectively), each following directly from their piecewise definitions; the OR-alternative instead fits a linear rule f(x)=ax+b to the given input-output pairs. …
Figure — The primary alternative asks to draw y=|x| and y= x ; the canonical modulus-function graph is an exact match,
f(x)=∣x∣ is a V-graph (two rays from the origin); f(x)=[x] (greatest integer / floor function) is a step graph made of unit-length horizontal segments. [OR alternative: for the linear function f(x)=ax+b fitted to the given points, a=2,b=−1.]
Part (i): f(x)=∣x∣.
By definition, ∣x∣=x for x≥0 and ∣x∣=−x for x<0.
So the graph consists of two rays meeting at the origin: the ray y=x for x≥0 (slope 1, through quadrant I) and the ray y=−x for x<0 (slope −1, through quadrant II). Together they form a V-shape with vertex at (0,0), symmetric about the y-axis, and f(x)≥0 always.
Part (ii): f(x)=[x] (greatest integer function). …
