Piecewise Functions— Evaluating, Graphing, Domain & Range Step by Step

In this video we cover Piecewise Functions from the ground up — what they are, how to evaluate them at any value of x, how to graph them correctly with open and closed circles, and how to find their domain and range. Every concept is taught step by step with fully worked examples drawn from university exam questions.

Piecewise functions are one of the most misunderstood topics in Pre-Calculus, yet they appear constantly in Calculus 1 and advanced mathematics. Master them here.

━━━━━━━━━━━━━━━━━━━━━━━━━━━━
WHAT YOU'LL LEARN IN THIS VIDEO
━━━━━━━━━━━━━━━━━━━━━━━━━━━━
✦ What a piecewise function is and how to read the notation
✦ How to evaluate a piecewise function at a given value of x
— identifying which piece applies
— substituting correctly into the right expression
✦ How to graph a piecewise function step by step:
— plotting each piece over its restricted domain
— open circles (excluded endpoints)
— closed circles (included endpoints)
— checking for continuity at the breakpoints
✦ Finding the domain of a piecewise function
✦ Finding the range of a piecewise function from its graph
✦ Worked example 1 — two-piece function (linear + quadratic)
✦ Worked example 2 — three-piece function with constants
✦ Worked example 3 — finding domain and range from a graph
✦ Common mistakes and how to avoid them
✦ Exam tips for piecewise function questions

━━━━━━━━━━━━━━━━━━━━━━━━━━━━
WHO THIS VIDEO IS FOR
━━━━━━━━━━━━━━━━━━━━━━━━━━━━
• Pre-Calculus and Calculus 1 students
• Anyone who finds piecewise functions confusing
• Students preparing for university mathematics examinations

━━━━━━━━━━━━━━━━━━━━━━━━━━━━
PREREQUISITES
━━━━━━━━━━━━━━━━━━━━━━━━━━━━
• Watch Part 1 — Functions (domain, range, types)
• Watch Part 2 — Domain and Range of a Function
• Basic graph plotting and linear equations

━━━━━━━━━━━━━━━━━━━━━━━━━━━━
TIMESTAMPS
━━━━━━━━━━━━━━━━━━━━━━━━━━━━
0:00 Introduction — what is a piecewise function?
2:00 Reading piecewise function notation
4:30 Evaluating — how to pick the right piece
7:00 Evaluating — worked example 1
10:00 Evaluating — worked example 2
13:00 Graphing — open vs closed circles explained
16:00 Graphing — step-by-step method
19:00 Graphing — worked example (two-piece)
23:00 Graphing — worked example (three-piece)
27:00 Domain of a piecewise function
30:00 Range of a piecewise function
33:00 Domain and range from a graph — worked example
36:00 Common mistakes and exam tips
38:30 Summary

━━━━━━━━━━━━━━━━━━━━━━━━━━━━
PRE-CALCULUS SERIES — PLAYLIST
━━━━━━━━━━━━━━━━━━━━━━━━━━━━
Part 1 → Functions — Domain, Range, Types & Composition
Part 2 → Domain and Range of a Function
Part 3 → Piecewise Functions (you are here)
More topics coming soon — subscribe so you don't miss it.

Full Pre-Calculus Playlist: [Add playlist link here]

━━━━━━━━━━━━━━━━━━━━━━━━━━━━
ALSO ON THIS CHANNEL
━━━━━━━━━━━━━━━━━━━━━━━━━━━━
📐 Algebra Series — Induction, Polynomials, Set Theory
📉 Coordinate Geometry — Parabola, Ellipse, Hyperbola, Circle
🔢 ODE Series — Exact, Non-Exact, Second Order ODEs

━━━━━━━━━━━━━━━━━━━━━━━━━━━━
ABOUT MATH WITH ALGEPHILE
━━━━━━━━━━━━━━━━━━━━━━━━━━━━
Math with Algephile is a university-level mathematics channel covering Pre-Calculus, Algebra, Calculus, Geometry, Number Theory, and Statistics. Every video is built around deep understanding — the why behind the how.

New videos every week. Subscribe so you never miss a topic.

━━━━━━━━━━━━━━━━━━━━━━━━━━━━
CONNECT
━━━━━━━━━━━━━━━━━━━━━━━━━━━━
📌 Subscribe: [Your YouTube Link]
📘 Instagram: @mathwithalgephile
🐦 X/Twitter: @algephile

#PiecewiseFunctions #PreCalculus #MathWithAlgephile

Видео Piecewise Functions— Evaluating, Graphing, Domain & Range Step by Step канала Math With Algephile