Composite and Inverse Functions
Composition of a function and its inverse are two mathematical concepts with practical applicability. The objective of these two concepts is to increase the understanding of functions and all the terms related to it. Students will better understand the definitions of domain and range after going through these two concepts.
The objective of the composition of functions and inverse of a function is to develop an application based thinking of how the functions work. Both of these concepts have a real-life application. Students are advised to regularly give time and effort to mathematics and increase their score in it.
Composite Functions and Inverse Functions
Let us try and understand both of these mathematical concepts in detail. Composition of functions and inverse functions are easy concepts to understand and apply. Students are advised to try as many examples as possible to solidify their learning and understanding of both the concepts. Given below is the detailed explanation of both the concepts:
Composition of Functions
Composition of function is defined when the result of a function is obtained by applying another function. The independent variable is another function. Let us try to understand the composition of functions with the help of an example.
Let there be two functions, f(x) and g(x).
f(x) = 2x + 1 g(x)=x2
Let us find the value of g(x) with the result obtained from f(x).
Calculate f(x) at 1.
f(1)=2.1+1=3
Let us calculate g(x) at 3.
g(3)=32= 9.
To streamline the above process and understand it better, we create a new function. This is how we represent the composition of functions.
f(g(x))=(fog)(x)
Where o is the composition operator and is used to define the composition of functions. Let us try and solve the above problem with this representation at x=1.
(gof)(x)=g(f(x))
=g(2x+1)
=(2x+1)2
=4x2+ 4x + 1
= 9
The notation (fog) is read as f of g or f is composed of g.
Inverse Functions
Inverse functions, as the name suggests, is to describe an inverse relationship between two functions. The two functions are opposite of each other. Let us try and understand this concept using a common example.
Let us take the case of temperature scales. Two scales, degree Celsius scale and the Fahrenheit scale, are used to measuring temperature.
C(x)=5/9(x-32) converts fahrenheit to degree celsius.
F(x)=9/5(x) + 32 converts degree celsius into degree fahrenheit.
We wish to convert 77oF into degree celsius.
C(x)=5/9(77-32)
=5/9(45)
=5.5
=25oC
Now to understand the inverse function, let us convert this degree celsius into degree fahrenheit.
F(x)=9/5(x)+32
=9/5(25)+32
=9.5+32
=45+32
=77oF
We get the same Fahrenheit value we began with. Inverse functions are opposite of each other. We can obtain any of the functions.
C(x) = 5/9(x-32)
9/5(C(x)) = x-32
9/5(x) + 32 = F(x)
This is how the inverse functions work.
FAQs on Introduction to the Composition of Functions and Inverse of a Function
1. What is meant by the composition of functions in mathematics?
The composition of functions is a process where the output of one function becomes the input for another. If f(x) and g(x) are two functions, their composition is represented as (fog)(x) = f(g(x)). This concept helps to create new functions and analyze how changes in one function affect the result of another, as required by the CBSE 2025–26 Maths syllabus.
2. How do you find the composition (fog)(x) and (gof)(x) for given functions?
To calculate (fog)(x): substitute g(x) into f(x). For (gof)(x): substitute f(x) into g(x).
- Example: If f(x) = 2x + 1 and g(x) = x2, (fog)(x) = f(g(x)) = 2(x2) + 1 = 2x2 + 1.
- (gof)(x) = g(f(x)) = (2x+1)2 = 4x2 + 4x + 1.
3. What is the inverse of a function and how can you determine if it exists?
An inverse function reverses the effect of a given function. A function has an inverse if and only if it is both one-to-one (injective) and onto (surjective). To find the inverse:
- Replace f(x) with y.
- Solve for x in terms of y.
- Replace y with x to express the inverse function as f-1(x).
4. Why is understanding the domain and range important when dealing with composite and inverse functions?
The domain and range dictate where a composition or inverse is defined. For composite functions, the output of the first function (g(x)) must fit within the domain of the second (f). For inverse functions, the original function's domain becomes the inverse's range and vice versa. Understanding these sets prevents undefined results and mistakes in problem-solving, especially in board exams.
5. How can inverse trigonometric functions be used in composition, and what restrictions should be kept in mind?
For inverse trigonometric functions, composition commonly follows identities like sin(sin-1(x)) = x, valid only when x lies in the principal domain (e.g., -1 ≤ x ≤ 1 for sine). When composing with different functions, remember that each trigonometric function and its inverse has a unique domain and range, ensuring only valid real values are included in exam solutions.
6. What are common errors students make with composite and inverse functions during CBSE board exams?
Frequent mistakes include:
- Incorrect substitution of one function into another (mixing up f(g(x)) with g(f(x))).
- Ignoring domain restrictions, leading to undefined answers.
- Assuming every function has an inverse without checking for one-to-one correspondence.
- Errors in solving for x while finding the inverse.
7. How does the composition of functions appear in real-life applications?
Composition of functions models scenarios where one process depends on the result of another, such as temperature conversion (Celsius to Fahrenheit), currency exchange rates, or layering mathematical transformations in computer graphics. These practical uses reflect why the concept is vital in the NCERT Maths syllabus.
8. Can every function have an inverse? Under what conditions is this possible?
Not all functions have inverses. A function admits an inverse only when it is bijective (both injective and surjective). For example, f(x) = x2 defined over the set of real numbers is not invertible as it is not one-to-one, but if restricted to x ≥ 0, it becomes invertible. Recognizing these conditions is essential for accurate board exam responses.
9. Explain with steps how to find the inverse of f(x) = x4 as per CBSE methods.
To find the inverse of f(x) = x4, follow these steps:
- Let y = x4.
- Solve for x: x = y1/4.
- Express as an inverse function: f-1(x) = x1/4, considering domain and range restrictions, such as x ≥ 0 for real results.
10. How does the topic "Introduction to the Composition of Functions and Inverse of a Function" contribute to higher-order thinking skills (HOTS) in mathematics?
This topic enhances higher-order thinking by requiring students to analyze multiple steps, identify dependencies between functions, manipulate domains, and predict complex behavior from simple mathematical rules. Mastery helps in approaching advanced mathematical problems systematically, reflecting the CBSE's emphasis on conceptual understanding.
