2nd Year Mathematics - Chapter 1 - New Book Exercise 1.3 - Complete solution

Students preparing for their second year of mathematics usually find Chapter 1 difficult, since functions involve more than just plugging numbers into formulas. This chapter introduces the graphical and mathematical concepts that are later used in calculus, especially when students begin studying limits, continuity, and derivatives. In the present list of Class 12 Mathematics resources, Chapter 1 is classified as being about functions and graphs, and Exercise 1.3 is given as a separate exercise in the new course material. The article gives a simple explanation of the key points relating to Exercise 1.3 of Chapter 1 in the new Second Year Mathematics book, with a special focus on one-to-one functions, the horizontal line test and the Key Transformation Rules of functions. Rather than viewing each rule as something to be memorized without question, it is better to understand how the graph changes as the function's equation changes. When this idea is clear, a number of questions that at first seem difficult become a lot easier to solve.

2nd Year Mathematics - Chapter 1 - New Book Exercise 1.3 - Complete solution

Introduction to Exercise 1.3

Exercise 1.3 is an important element of Chapter 1 since it helps to reinforce the student’s understanding of functions and how their graphs behave. The latest 2026 educational material outlines the new content for Class 12 Exercise 1.3 on functions and transformations, covering topics such as inverse functions, modulus functions, and shifting and scaling. A student should therefore pay attention not just to getting the right answer but also to understanding why a given transformation affects the graph in a particular way. You can think of a function as being like a machine: something enters the machine, a rule acts on it, and exactly one output comes out. If we alter the rule, we might move, stretch, shrink, or reflect the graph, but the fundamental relationship between input and output still remains at the heart of the matter. This way of looking at things makes the exercise more logical and helps students avoid memorising formulas without understanding what they mean.

Main Concepts Covered in the Exercise

The topics involved in this exercise are functions, one-to-one functions, inverse relationships and transformations of graphs. They are especially useful since they link algebra with geometry. For instance, if we write y = f(x), we can study the function either by using its formula in an algebraic way or by referring to its graph visually. A transformation enables us to start with a known graph and create a new one without having to plot all the points from scratch. Moreover, if we know that a function is one-to-one we can determine whether its inverse will also be a function. The present resources for Class 12 show that Exercise 1.3 is included in the larger Functions and Graphs section of the course.

Understanding One-to-One Functions

A function is said to be one-to-one when different elements from its domain give rise to different elements in its range. Put simply, two distinct inputs must not result in the same output. Mathematically, if f(a) = f(b) then a one-to-one function demands . This characteristic is very important when considering inverse functions since the inverse has to be able to determine the original input from the output. If more than one different input leads to the same output, the inverse would not know which input to select. Because of this, one-to-one functions occupy an important position in the study of inverse functions and graph transformations.

One-to-One Using the Horizontal Line Test

The horizontal line test offers a simple graphical method for determining if a function is one-to-one; suppose you draw horizontal lines across the graph, in that case if any such horizontal line cuts the graph at more than one point the function is not one-to-one, but if every horizontal line intersects the graph at most once then the function is one-to-one.

A horizontal line like cuts the graph at and it follows that , when its domain is all real numbers, is not one-to-one.

It is especially useful in the context of exam questions since it enables you to judge directly from the graph whether or not a function is one-to-one rather than having to depend completely on algebraic calculations.

Key Transformation Rules of Functions

The key transformation rules for functions are some of the most useful points to remember from this topic. Let us assume that the original function is:

The graph shows predictable changes when different modifications of this expression are made.

Transformation	New Function	Effect on Graph
Vertical shift upward		Moves up units
Vertical shift downward		Moves down units
Horizontal shift right		Moves right units
Horizontal shift left		Moves left units
Reflection in x-axis		Flips vertically
Reflection in y-axis		Flips horizontally
Vertical stretch	,	Stretches away from x-axis
Vertical compression	,	Compresses toward x-axis

A common mistake is to mix up horizontal and vertical transformations. If the change is outside the function, the graph is moved vertically; but if the change is inside the function, the graph is moved horizontally. The signs may also be the opposite of what students first expect. For instance, * shifts the graph 3 units to the right, while * shifts it 3 units to the left. The best approach is to remember that modifications inside the brackets affect the x-direction, whereas those outside affect the y-direction.

These examples illustrate the importance of understanding function transformations by looking at graphs. There's no need to calculate numerous points each time; if you can identify the original graph and know the transformation rule, you will be able to draw the new one much more efficiently.

A step-by-step method for solving Exercise 1.3

In Exercise 1.3, the very first thing you should do when attempting a question is to make certain you know exactly what is being asked. For example, are you being asked to decide if a function is one-to-one? Are you required to carry out a graph transformation? Or are you being asked to find the inverse or to describe the effect of a change in the equation? Once you have made clear what is required, write down the original function and then compare it with the new one. You must carefully examine the changes both inside and outside the brackets since it is precisely these that show the kind of transformation and its direction.

When dealing with graphical questions, start with the original graph if you can. Identify a number of key points, for example the intercepts, the turning points or the endpoints. Then apply the transformation to these points rather than drawing the graph from scratch. In the case of one-to-one questions, use the horizontal line test or give an algebraic reason. If you are verifying whether implies , simplify both sides carefully and do not cancel out expressions without first considering their possible values. Adopting this systematic method makes the solution easier to follow and lowers the chance of losing marks because of a minor algebraic error.

Important Exam Tips and Common Mistakes

Students usually have marks deducted in this chapter since they mix up the direction of horizontal shifts. Remember:

A frequently made error is to carry out the horizontal line test in the wrong way. The vertical line test is used to determine if a graph represents a function, whereas the horizontal line test is used to see if a function is one-to-one. Once again, it is essential to tell the two tests apart: the first one reflects the graph in the y-axis, while the second one reflects it in the x-axis. The rules could be put into a small revision table, which would save a lot of time when preparing for the board examination.

Conclusion

When the functions are treated as both algebraic expressions and as graphs, the 2nd Year Mathematics New Book Chapter 1 Exercise 1.3 becomes a lot easier. The main points involve identifying one-to-one functions, using the horizontal line test and understanding how equations affect graphs. The key transformation rules of functions offer a very effective shortcut since they enable students to predict movements, reflections, and scaling without having to draw the whole graph each time. The latest 2026 resources still include Exercise 1.3 in the section on Class 12 functions and graphs, so these ideas remain relevant for students who are following the revised curriculum. The best way to prepare is to understand the rule, do a number of examples, draw the graphs carefully and always make sure that the final result agrees with the transformation mentioned in the question.

Frequently Asked Questions

1. What counts as a one-to-one function?

A function is one-to-one when each output is associated with just one input. If , then . This property is important since a one-to-one function can have an inverse function over its relevant domain and range.

2. How does the horizontal line test identify a one-to-one function?
Imagine that you draw some horizontal lines across the graph. The function is not one-to-one if any of these horizontal lines cut the graph more than once. However, the function will pass the horizontal line test if each horizontal line intersects at only one point.

3. What does -f(x) do to a graph?

It flips the graph of f(x) over the x-axis. Each point (x, y) on the graph of f(x) moves to (x, -y) on the graph of -f(x). The transformation f(-x) reflects the graph across the x-axis; each y-coordinate changes sign while the x-coordinate stays the same.

4. Why are function transformations important in Class 12 Maths?
Function transformations allow students to quickly understand graphs and provide a basis for future topics such as inverse functions, limits, and calculus. Rather than having to plot each graph from scratch, students can use a small number of dependable rules to transform a graph that they are already familiar with.