Introduction

Following our Unit 1 notes on Complex Numbers, the Unit 2 solved notes for Functions and Graphs are now ready for Class 11 students across all nine Punjab boards.

This unit brings together algebra and visual thinking, and once you’re comfortable with it, a lot of what comes later in the syllabus starts to feel a lot easier. These notes are based on the new PECTAA book and are written the same friendly, step by step way our students already know from Unit 1, so you can move between units without missing a beat.

Built for Every Punjab Board, and Every Kind of Learner

Whether you’re prepping for board exams at a cadet college, working through entry test material at a private school, or just trying to keep pace with a fast-moving classroom, this unit is written to meet you where you are. We keep the explanations plain, the diagrams clear, and every solved question fully shown so nothing is left for you to guess.

What’s Inside Section 2.1, Understanding Functions

Section 2.1 lays the groundwork for the whole unit. It starts with the basic idea of a function as a rule that connects an input to exactly one output, then walks through domain, range, and codomain, and how they relate to each other. From there it covers function notation, how to evaluate a function at a given value or expression, and the vertical line test, a quick visual way to check whether a graph actually represents a function.

One of the worked examples in this section evaluates the function f(x) = x cubed minus 2x squared plus 4x minus 1 at several inputs, including plain numbers, a shifted expression like 1 plus x, and even a reciprocal input like 1 over x. Seeing the same function handled in so many different ways in one place is a great way to build real comfort with function notation before moving on.

The section then covers the three big types of function you’ll be tested on: one-to-one (injective), onto (surjective), and bijective functions, along with a look at piecewise functions, where a function switches between different rules depending on which part of the domain you’re in. Exercise 2.1 is fully solved, covering domain and range questions, difference quotient problems, expressing one quantity as a function of another, and proving whether a given function is one-to-one, onto, or bijective.

Section 2.2, Finding Where Graphs Meet

This is where functions and graphs really come together. Section 2.2 shows you how to find the point, or points, where two graphs intersect, purely by graphing them and reading off the coordinates. It’s broken into three parts: a linear function meeting the coordinate axes, two linear functions meeting each other, and a linear function meeting a quadratic function.

The linear-and-quadratic case is the trickiest of the three, since a line and a parabola can cross at two points, touch at exactly one point, or miss each other completely, and the notes walk through how to tell which situation you’re in using the discriminant. Exercise 2.2 is fully solved, taking you through every combination: axes with lines, line with line, and line with parabola, so you build confidence reading and sketching intersections rather than just calculating them.

Sections 2.3 and 2.4, Square Root and Cube Root Graphs

These two sections focus on sketching and interpreting square root and cube root functions. You’ll learn how to find the domain and range of these functions before you even plot them, then use a table of values to draw the curve and see how shifts and stretches change its shape. The cube root section is especially useful for understanding how a graph can pass smoothly through a point where a square root graph simply couldn’t exist, which is a common source of confusion at this stage.

Section 2.5, Real Life Applications

This is the section students tend to enjoy most, because it shows exactly why this unit matters outside the exam hall. It covers exponential growth and decay, including how to model a stock price growing at a fixed yearly rate and work out how long it takes to double in value, and how to model something like a pollutant in a lake gradually decaying over time and predict when it drops below a safe threshold. These real life examples make the algebra feel a lot less abstract.

Bonus Resources

• A Functions and Graphs formula sheet PDF for quick revision.
• Quick tips for spotting one-to-one, onto, and bijective functions without long proofs.
• Functions and Graphs video lectures.
• Unit 2 past papers solved, covering the years 2018 through 2025.
• Unit 3 is coming soon as the next unit in this series.

View the Full Solved PDF

The complete solved PDF for Unit 2 is available to view for free.

Why This Unit Matters for Your Overall Score

Functions and Graphs sits right after Complex Numbers for a reason. The domain and range work in Section 2.1 comes up again and again in later units, and the graphical intersection skills from Section 2.2 are exactly what you’ll need when you get to more advanced curve sketching down the line.

Students who take the time to really understand this unit, rather than memorizing the exercise answers, tend to find the rest of the Mathematics syllabus much more manageable.

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Frequently Asked Questions

What topics does Unit 2 cover? Functions and their types, function notation and evaluation, the vertical line test, graphical intersections of linear and quadratic functions, square root and cube root graphs, and real life growth and decay applications.

How do I know if a function is one-to-one, onto, or bijective? A function is one-to-one if no two different inputs give the same output, onto if every element of the codomain is actually produced by some input, and bijective if it’s both at once. Section 2.1 walks through the tests for each with worked examples.

Do the notes include past papers? Yes, Unit 2 past papers solved from 2018 through 2025 are included as a bonus resource, just like with Unit 1.

Do I need to download the PDF to view these notes? No, the full solved PDF can be viewed directly on the site for free, with no download required.

Conclusion

With Sections 2.1 through 2.5 and both exercises fully solved, Class 11 students across all nine Punjab boards now have a complete, exam ready resource for Functions and Graphs. Pair these notes with the bonus formula sheet and past papers to keep building the same momentum you started with Unit 1.

Best of luck for your Class 11 Mathematics exam!