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 Signup and view all the answers

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

It is a bijection 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 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 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 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 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 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 Signup and view all the answers

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

'Onto' property 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 Signup and view all the answers

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

f2 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 Signup and view all the answers

Which function from the given figures is onto?

f4 Signup and view all the answers

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

Invertible function 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 Signup and view all the answers