Invertible and non invertible function

1. What is a non invertible function?
2. What is invertible function?
3. What is the difference between inverse and invertible function?
4. How do you show that a function is not invertible?

What is a non invertible function?

The inverse of a function is not necessarily a function. 𝑦 = 𝑥², for example, because as we invert it we get 𝑥 = ±√𝑦, so each positive 𝑦-value is now mapped to two different 𝑥-values. which means that 𝑥 is not a function of 𝑦 and we say that 𝑦 = 𝑥² is non-invertible.

What is invertible function?

Invertible function

A function is said to be invertible when it has an inverse. It is represented by f−1. Condition for a function to have a well-defined inverse is that it be one-to-one and Onto or simply bijective. Example : f(x)=2x+11 is invertible since it is one-one and Onto or Bijective.

What is the difference between inverse and invertible function?

["]A function is invertible if and only if it is injective[."] So for a function to have a inverse, it must be bijective. But any function that is injective is invertible, as long as such inverse defined on a subset of the codomain of original one, i.e. the image of the original function?

How do you show that a function is not invertible?

Using the second derivative test, we can state this condition in terms of derivatives: if f′(x0)=0 and f″(x0)≠0, then f fails to be locally invertible at x0. However, if f″(x0)=0, the second derivative test fails, and f may or may not be locally invertible (as the example f(x)=x3 given in the comments shows).