Class 10 Maths Chapter 4 Exercise 4.1 Solutions 2026 – Functions and Graphs Notes PDF

Unit 4: Functions and Graphs | Exercise 4.1 | Punjab Board New Syllabus 2026–27

Updated August 2026: Exercise 4.1 solutions for Functions and Graphs have been uploaded below and are free to view or download, fully solved according to the new PCTB/PECTAA Class 10 Mathematics syllabus.

Giving Every Input Exactly One Output

A function is one of the most fundamental ideas in all of mathematics: a rule that takes an input and always produces exactly one output for it. Exercise 4.1 introduces this idea formally, along with the vocabulary — domain, range, and the different types of functions — needed to describe it precisely.

Key Concepts Covered in Exercise 4.1

What a Function Is

A function f from a set A to a set B assigns to each element of A exactly one element of B. The key word is “exactly one” — if a single input could produce two different outputs, the rule would not be a function at all.

Domain and Range

The domain is the complete set of allowed input values (the x-values), and the range is the set of all actual output values produced (the y-values). For a fraction like f(x) = 1/x, the domain excludes x = 0, since division by zero is undefined.

The Vertical Line Test

On a graph, a relation is a function only if every vertical line crosses the graph at most once. If any vertical line touches the graph twice, that x-value would have two different y-values, breaking the definition of a function.

Types of Functions

This exercise introduces several important types: linear functions (f(x) = mx + c, whose graph is a straight line), quadratic functions (f(x) = ax² + bx + c, whose graph is a parabola), and constant functions (f(x) = c, whose graph is a horizontal line).

Step-by-Step Solved Examples

Q. No. 1: Find the domain of f(x) = 1/(x − 3).

The function is undefined when the denominator equals 0.

x − 3 = 0  →  x = 3

Domain: all real numbers except x = 3

Q. No. 2: If f(x) = 2x + 1, find f(3) and f(−2).

f(3) = 2(3) + 1 = 6 + 1 = 7

f(−2) = 2(−2) + 1 = −4 + 1 = −3

Q. No. 3: Is the relation {(1, 2), (2, 3), (1, 5)} a function? Explain.

Check each input: the input 1 appears twice, mapping to both 2 and 5.

Since one input (1) produces two different outputs (2 and 5), this is NOT a function.

These three questions are representative samples. The full PDF notes uploaded on this page walk through every question of Exercise 4.1 in the same numbered, step-by-step format.

MCQs, Short Questions & Long Questions from This Exercise

Alongside the main exercise, this page’s uploaded notes are organized the way Punjab Board papers are structured, so students can practice by question type:

• MCQs: Quick questions on identifying the domain of a simple function or applying the vertical line test
• Short Questions: Evaluating a given function at specific input values, like Q. No. 2 above
• Long Questions: Determining whether a given relation or graph represents a function, with justification

Where This Matters Beyond This Exercise

Domain and range become essential again in the next exercise when sketching linear and quadratic graphs, since knowing which x-values are valid determines exactly what can be plotted. Functions are also the mathematical backbone of nearly every applied field — physics formulas, economic models, and computer programs are all, at their core, functions mapping inputs to outputs.

Common Mistakes Students Make in Exercise 4.1

• Forgetting to exclude values that make a denominator zero when finding the domain
• Confusing the domain (inputs) with the range (outputs)
• Assuming any pair of related x and y values automatically forms a function, without checking for repeated inputs
• Misapplying the vertical line test by checking horizontal lines instead

Why This Exercise Matters for the Board Exam

Domain-finding questions are short, formulaic, and highly repeatable — usually just a case of setting a denominator or expression under a square root to avoid an invalid value. Function evaluation questions like f(3) are equally quick, making Exercise 4.1 one of the most time-efficient sources of marks in this chapter.

Quick Links – Chapter 4: Functions and Graphs

Links for sections other than Exercise 4.1 will be activated as they are published on this site.

Download Exercise 4.1 Notes PDF

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Frequently Asked Questions (FAQs)

Q1. Are these Exercise 4.1 notes free to download?

Yes, all notes on this page are completely free to view and download in PDF format.

Q2. Which board are these notes for?

These notes are prepared according to the Punjab Board syllabus and are useful for all Punjab boards (Lahore, Gujranwala, Multan, Sargodha, Rawalpindi, Faisalabad, DG Khan, Bahawalpur, Sahiwal) as well as the Federal Board (FBISE).

Q3. Do I need to know earlier chapters before starting Exercise 4.1?

No — Chapter 4 (Functions and Graphs) is independent of Chapters 1 to 3. Exercise 4.1 introduces functions completely from scratch.

Q4. Does this page include MCQs and short questions too?

Yes, the uploaded notes include MCQs, short questions, and long questions related to this exercise’s topics, organized by question type to match the board paper pattern.

Q5. How can I download the PDF?

Click the “Download PDF” button above and the notes will open or download directly to your device.

Q6. Are these notes updated for the new 2026–27 syllabus?

Yes, these notes are prepared strictly according to the latest PCTB/PECTAA syllabus for the 2026–27 academic session.

Comments & Feedback

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