CALCULUS: COMPOSITION OF FUNCTIONS (CBSE CLASS XII 12th Maths)

Summary

In this video, Neha from Mathematically Inclined explains the composition of functions, a key topic for Class 12 Mathematics and competitive exams. She breaks down the concept with clear examples, showing how functions can be combined and how their domains and ranges affect the composition. Through real-world analogies and step-by-step walkthroughs, viewers learn how to compute compositions like f(g(x)) and g(f(x)). The video also covers self-composition and offers practical insights to simplify complex mathematical expressions. A must-watch for students looking to master function composition!

Takeaways

• 😀 Composition of functions involves combining two functions, where the output of one function becomes the input of the other.
• 😀 If f is a function from set A to set B, and g is from B to C, the composition g(f(x)) defines a function from A to C.
• 😀 The composition of functions is like a chain reaction where one function's output serves as the input for the next.
• 😀 For g(f(x)) to exist, the range of f must match the domain of g. Similarly, for f(g(x)) to exist, the range of g should be a subset of the domain of f.
• 😀 The order of composition matters: g(f(x)) and f(g(x)) are not the same and yield different results.
• 😀 Example 1 shows that for f(x) = x² and g(x) = 2x + 1, f(g(x)) results in (2x + 1)², while g(f(x)) results in 2x² + 1.
• 😀 Composition functions can be expressed in terms of ordered pairs, with the domain representing the first coordinates and the range representing the second.
• 😀 When working with ordered pairs, calculating f(g(x)) and g(f(x)) involves applying the values from the domain and range to find the respective results.
• 😀 Example 2 demonstrates how to compute f(g(x)) and g(f(x)) using given functions expressed in ordered pairs, leading to specific outcomes.
• 😀 The self-composition of a function, like f(x) = x² - 3x + 2, involves substituting the function into itself and simplifying the result, leading to more complex expressions.
• 😀 Understanding composition of functions is important not only for CBSE exams but also for competitive exams, as it helps in problem-solving and functional analysis.

Q & A

What is the basic concept behind the composition of functions?

The composition of functions involves applying one function to the output of another. For example, if you have a function F from A to B and another function G from B to C, applying F to an input X results in F(X), and applying G to the output of F(X) gives G(F(X)). This forms a new function that maps from A to C.

What does the notation G of F(x) represent?

The notation G of F(x) means that the function F is applied first, and then the function G is applied to the result of F(x). This represents the composition of the two functions where G is applied to the output of F.

How do you define the composition of functions?

If F is a function from A to B, and G is a function from B to C, the composition of functions G and F, written as G ∘ F, is a function from A to C, defined as G(F(x)).

What is the relationship between the range of F and the domain of G for the composition G ∘ F to exist?

For the composition G ∘ F to exist, the range of F must be a subset of the domain of G. This ensures that every output from F can be an input to G.

What is the key difference between F of G and G of F?

The key difference is the order in which the functions are applied. For F of G, you apply G first and then F. For G of F, you apply F first and then G.

In the example with the functions F(x) = x^2 and G(x) = 2x + 1, what is F(G(x))?

For F(G(x)), you substitute G(x) into F. G(x) = 2x + 1, and F(x) = x^2. So, F(G(x)) = (2x + 1)^2.

In the same example, what is G(F(x))?

For G(F(x)), you substitute F(x) into G. F(x) = x^2, and G(x) = 2x + 1. So, G(F(x)) = 2(x^2) + 1.

How do you find the composition of functions when given functions in the form of ordered pairs?

When functions are given as ordered pairs, first identify the domain and range of each function. Then, for each element in the domain of one function, apply the second function and use the result to find the output from the first function.

In the example with ordered pairs, what is the result of F(G(2))?

In this example, G(2) = 3, and then applying F to 3 gives F(3) = 5. So, F(G(2)) = 5.

What is the result of G(F(1)) in the example with ordered pairs?

F(1) = 2, and G(2) = 3. Therefore, G(F(1)) = 3.