Invertible Functions Definition and Properties

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Which property ensures that a function is invertible?

Being one-one and onto (correct)

Reflexivity

Symmetry

Being continuous

What is the definition of an invertible function?

A function with a unique value for each input

A function that is reflexive and symmetric

A function that is one-one and onto (correct)

A function that is continuous and increasing

In the context of functions, what does the term 'inverse' refer to?

A transformation of the original function

The opposite direction of the function

A function that undoes the original function (correct)

The reciprocal of the function

Why is it beneficial to prove a function is one-one and onto when determining its invertibility?

To avoid the need to calculate the actual inverse function (B) Signup and view all the answers

If a function f : X → Y is invertible, what can be concluded about f?

It is a bijection (B) Signup and view all the answers

How can proving a function to be one-one and onto help establish its invertibility?

By ensuring there are no repeated outputs in the codomain (D) Signup and view all the answers

For the function f(x) = x^4, is f one-one and onto?

No, f is many-one onto (A) Signup and view all the answers

For the function f(x) = 3x, is f one-one and onto?

No, f is one-one but not onto (A) Signup and view all the answers

If f : A → B and g : B → C are functions, what does the composition gof represent?

Application of g after applying f (A) Signup and view all the answers

In the context of functions, what does it mean for a function to be 'many-one'?

Each element in the domain maps to multiple elements in the codomain (D) Signup and view all the answers

What does it mean for a function to be 'onto'?

Each element in the codomain is mapped to by at least one element in the domain (B) Signup and view all the answers

When proving invertibility of a function, what property must the function satisfy?

'Onto' property (A) Signup and view all the answers

What is the defining characteristic of a one-one function?

The images of distinct elements of X under f are distinct (A) Signup and view all the answers

Which of the following functions is NOT one-one based on the given information?

f2 (C) Signup and view all the answers

What does an onto function ensure according to the text?

Every element in the codomain has a pre-image in the domain (C) Signup and view all the answers

Which function from the given figures is onto?

f4 (C) Signup and view all the answers

A function that is both one-one and onto is known as:

Invertible function (D) Signup and view all the answers

What is the defining characteristic of a many-one function?

The images of distinct elements of X can be equal (D) Signup and view all the answers

Invertible Function

A function that is both one-to-one (or injective) and onto (or surjective).

One-to-one (Injective)

A function where each element in the domain maps to a unique element in the codomain.

Onto (Surjective)

A function where every element in the codomain has at least one corresponding element in the domain.