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Invertible And Composite Functions

MathsFunctionsFor JEE aspirants

A function is invertible if and only if it is bijective (one-one and onto). Its inverse satisfies and , and its graph is the reflection of the graph of across the line . The composite function applies first, then , and is defined when the range of lies inside the domain of . In general .

Key Formulas - Quick Reference
  1. Inverse exists is one-one and onto (bijective)
  2. (identity)
  3. Domain Range; Range Domain
  4. Graphs of and are symmetric about
  5. Roots of lie on the line
  6. Composite: ; defined when Range Domain
  7. In general (composition is not commutative)

1. Inverse Function

Let be a function defined by . If is both one-one and onto, there exists a unique function such that for each , if and only if . This is called the inverse of and denoted .

Key idea: If , then . The inverse reverses every input-output pair of .

Important Properties of the Inverse

  • The inverse of exists if and only if is one-one and onto (bijective).
  • Domain of Range of ; Range of Domain of .
  • (composition with inverse is the identity).
  • The graphs of and are mirror images across the line .
  • Any root of the equation lies on the line (that is, ).
Inverse function as reverse mapping For a bijective function f from X to Y, the inverse f-inverse maps each element of Y back to its unique pre-image in X. Solid arrows show f mapping a to a1, b to b1, c to c1; dotted arrows show f-inverse mapping a1 back to a, b1 to b, c1 to c. X Y a b c a₁ b₁ c₁ f f⁻¹
Figure 1: The inverse function f⁻¹ reverses every input-output pair of f

How to Find the Inverse

  1. Check that is one-one (monotonic or algebraic test).
  2. Check that is onto (range equals co-domain).
  3. Write and solve for in terms of .
  4. Swap the roles: replace with to write .
Solved Example 1
The function is defined by . Find .
Solution:

Step 1 - Check one-one: Take : . Differentiating: , so .

For , and , so . Hence is strictly increasing on , so it is one-one.

Step 2 - Check onto: and as . So range co-domain. Hence is onto.

Step 3 - Solve for the inverse: Let . Take : .

So . Using the quadratic formula:

.

Since we need , so take the sign:

.

Replacing with gives:

, defined for .

2. Composite Function

If and are two functions, the composite function is defined by

To compute , first take , then apply to it to get .

Composite function g of f from X to Z through Y The composite function (g of f) sends each x in X first to f(x) in Y and then to g(f(x)) in Z. Two arrows in sequence produce one combined mapping from X to Z. X Y Z x f(x) g(f(x)) f g g ∘ f
Figure 2: Composite function (g ∘ f)(x) = g(f(x)) - two mappings applied in sequence

Important Properties of Composition

  • Domain of : Domain and Domain.
  • Range of is a subset of Range; they need not be equal.
  • Similarly, with Domain Domain and Domain.
  • In general . Composition is not commutative.
  • If and are both one-one, then is one-one.
  • If and are both onto, then is onto.
  • If is bijective, then and .
Solved Example 2
Let and . Find and , and check whether they are equal.
Solution:

.

.

Clearly , confirming composition is not commutative.

Solved Example 3
If and , find the domain of .
Solution:

Domain of : all real . Range of : .

Domain of : .

For to be defined, we need , i.e. , which is always true.

Domain , and .

Common Mistakes to Avoid

Watch out
  • Trying to find the inverse of a many-one function. The inverse only exists when is bijective. Always verify one-one and onto first.
  • Confusing with . The inverse function is not the reciprocal.
  • Assuming . Composition is generally not commutative.
  • Forgetting to check that Range Domain before writing . Without this, the composite may not be defined for all in Domain.
  • Picking the wrong sign after applying the quadratic formula when solving for the inverse. Use the range of the original function to select the correct branch.

Frequently Asked Questions

Q1. When does the inverse of a function exist?

The inverse exists if and only if is bijective, meaning both one-one (injective) and onto (surjective). If fails either test, the inverse is not a well-defined function on the whole co-domain.

Q2. How are the graphs of and related?

The graph of is the reflection of the graph of across the line . So if lies on , then lies on .

Q3. What is the domain of the inverse function?

The domain of is the range of , and the range of is the domain of . Domain and range swap when you invert.

Q4. Is the same as ?

Not in general. Composition of functions is not commutative. For example, if and , then while , which are different.

Q5. How do I find the domain of a composite function?

For the domain is is in the domain of , and is in the domain of . Start with the domain of , apply , and keep only those inputs whose images land in the domain of .

Q6. If is one-one but not onto, does it have an inverse?

Not as a function with the original co-domain. But if you restrict the co-domain to the range of , then becomes bijective and the inverse exists. In practice, JEE problems often use this restriction.

Q7. What is the relation between and ?

Since the graphs of and are mirror images across the line , they can only intersect on that line. So every root of satisfies .

Q8. Are composition and inversion related?

Yes. If is bijective, then and . Also, if and are both bijective, then . Note the order reverses.

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