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Composition of Functions and Compatibility Conditions

For functions f: A to B and g: B to C, the composition g∘f is defined by (g∘f)(x) = g(f(x)) for x in A. The output of f must be suitable as an input for g.

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Student-friendly explanation

Composition means applying one function after another in the correct order. In g∘f, f acts first and g acts second. The order matters; in general, g∘f and f∘g are not the same. Composition is associative when the functions are compatible, but it is not generally commutative.

How to write this in exams

  1. 1

    Start with the exact idea

    For functions f: A to B and g: B to C, the composition g∘f is defined by (g∘f)(x) = g(f(x)) for x in A. The output of f must be suitable as an input for g.

  2. 2

    Then show how to use it

    1. Identify which composition is asked. 2. Start with the inner function on the right. 3. Substitute the full expression into the outer function. 4. Simplify carefully. 5. Check domain and codomain compatibility if functions are defined between sets rather than only by formulas.

  3. 3

    Add one concrete example

    If f(x) = x + 2 and g(x) = x^2, then (g∘f)(x) = g(x + 2) = (x + 2)^2, while (f∘g)(x) = f(x^2) = x^2 + 2. These are not equal in general.

  4. 4

    Avoid this incomplete answer

    Writing (g∘f)(x) = f(g(x)) because the symbols are read left to right instead of applying the rightmost function first.

Definition

For functions f: A to B and g: B to C, the composition g∘f is defined by (g∘f)(x) = g(f(x)) for x in A. The output of f must be suitable as an input for g.

Example

If f(x) = x + 2 and g(x) = x^2, then (g∘f)(x) = g(x + 2) = (x + 2)^2, while (f∘g)(x) = f(x^2) = x^2 + 2. These are not equal in general.

Rule to remember

Composition rule: (g∘f)(x) = g(f(x)). Compatibility condition: if f: A to B and g: B to C, then g∘f: A to C. Associativity: h∘(g∘f) = (h∘g)∘f when all compositions are defined. In general, g∘f is not equal to f∘g.

Sequence to remember

Composition rule: (g∘f)(x) = g(f(x)). Compatibility condition: if f: A to B and g: B to C, then g∘f: A to C. Associativity: h∘(g∘f) = (h∘g)∘f when all compositions are defined. In general, g∘f is not equal to f∘g. Arrow diagrams may be used: A --f->
B --g->
C. They help students see the direction of application and the required compatibility. 1. Identify which composition is asked. 2. Start with the inner function on the right. 3. Substitute the full expression into the outer function. 4. Simplify carefully. 5. Check domain and codomain compatibility if functions are defined between sets rather than only by formulas

Memory hook

In g∘f, the function nearest x works first.

Examples and method

Worked example

Let f: R to R be f(x) = 3x + 1 and g: R to R be g(x) = x^2 - 2. Then (g∘f)(x) = g(f(x)) = g(3x + 1) = (3x + 1)^2 - 2 = 9x^2 + 6x + 1 - 2 = 9x^2 + 6x - 1. Also (f∘g)(x) = f(x^2 - 2) = 3(x^2 - 2) + 1 = 3x^2 - 5. Therefore the two compositions are different.

Method to apply

1. Identify which composition is asked. 2. Start with the inner function on the right. 3. Substitute the full expression into the outer function. 4. Simplify carefully. 5. Check domain and codomain compatibility if functions are defined between sets rather than only by formulas.

Diagram support

Arrow diagrams may be used: A --f--> B --g--> C. They help students see the direction of application and the required compatibility.

How CBSE asks it

Exam questions ask students to compute g∘f and f∘g, find a missing function from a composition, or verify associativity for given functions.

Avoid common mistakes

Common confusion

Students often reverse the order and compute f(g(x)) when the question asks for g(f(x)).

Common wrong answer

Writing (g∘f)(x) = f(g(x)) because the symbols are read left to right instead of applying the rightmost function first.

Exam tip

Read g∘f from right to left for calculation: first f, then g. Also check whether the range of the first function fits the domain of the second.

Quick check

If f(x) = 2x - 1 and g(x) = x^2 + 3, find (g∘f)(x).

(g∘f)(x) = g(2x - 1) = (2x - 1)^2 + 3 = 4x^2 - 4x + 4.

Answer writing and exam use

1-mark answer

For functions f: A to B and g: B to C, the composition g∘f is defined by (g∘f)(x) = g(f(x)) for x in A. The output of f must be suitable as an input for g.

2-mark answer

For functions f: A to B and g: B to C, the composition g∘f is defined by (g∘f)(x) = g(f(x)) for x in A. The output of f must be suitable as an input for g. Composition rule: (g∘f)(x) = g(f(x)). Compatibility condition: if f: A to B and g: B to C, then g∘f: A to C. Associativity: h∘(g∘f) = (h∘g)∘f when all compositions are defined. In general, g∘f is not equal to f∘g. If f(x) = x + 2 and g(x) = x^2, then (g∘f)(x) = g(x + 2) = (x + 2)^2, while (f∘g)(x) = f(x^2) = x^2 + 2. These are not equal in general.

3-mark answer

Composition means applying one function after another in the correct order. In g∘f, f acts first and g acts second. The order matters; in general, g∘f and f∘g are not the same. Composition is associative when the functions are compatible, but it is not generally commutative. Composition rule: (g∘f)(x) = g(f(x)). Compatibility condition: if f: A to B and g: B to C, then g∘f: A to C. Associativity: h∘(g∘f) = (h∘g)∘f when all compositions are defined. In general, g∘f is not equal to f∘g. Let f: R to R be f(x) = 3x + 1 and g: R to R be g(x) = x^2 - 2. Then (g∘f)(x) = g(f(x)) = g(3x + 1) = (3x + 1)^2 - 2 = 9x^2 + 6x + 1 - 2 = 9x^2 + 6x - 1. Also (f∘g)(x) = f(x^2 - 2) = 3(x^2 - 2) + 1 = 3x^2 - 5. Therefore the two compositions are different. Exam questions ask students to compute g∘f and f∘g, find a missing function from a composition, or verify associativity for given functions. Writing (g∘f)(x) = f(g(x)) because the symbols are read left to right instead of applying the rightmost function first.
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