How simple do you want it to be?

https://www.khanacademy.org/math/precalculus/invertible/a/intro-to-invertible-functions

Basically, for a function to be invertible (which means you can invert it to get the inverse function), it has to be one-to-one (also known as an injective function)

This means that one input has to correspond to exactly 1 unique output.

So when inverted, that output now becomes the input, which will correspond exactly to 1 output (which was the input before you did the inversion)

So, for f(x) = x + 2, we know that it is the equation of a straight line.

So every input has exactly 1 unique output.

To find the inverse, make x the subject

x = f(x) - 2

Then replace x with f-¹(x) and f(x) with x

So the inverse has equation :

f-¹(x) = x - 2

This is basically mirroring the function by reflecting it about the line y = x.

Remember that a function and its inverse and mirror images of each other.

Any point (a,b) on the function will mirror to (b,a) on the inverse.

This is my answer: function f is invertible. Firstly, it is injective since every element in its domain points to each individual elements in its co-domain with no elements being mapped more than once by the elements in its domain. Lastly, it is surjective since every element in its co-domain have at least one mapped by elements in its domain. However, function g is not invertible since it does not meet the requirements of an injective function. For example, g(-1) = (-1)^2 = 1 is equal to g(1)= 1^2 = 1.

But for g(x) = x², we can't do this.


Because, every output (except g(x) = 0) results from more than 1 input (to be exact, from 2 inputs)

Or in other words, only x = 0 has exactly 1 unique output.

Eg.

When x = 0, g(x) = 0² = 0
When x = 2 or x = -2, g(x) = 4
When x = 5 or x = -5, g(x) = 25

You have 2 inputs corresponding to every output other than 0.


So this is not an injective function.


When you try to invert it,


x = ± √g(x)

Replace x with g-¹(x) and g(x) with x,

g-¹(x) = ±√x

Recall that a function has to pass the vertical test (there can only be 1 dependent variable corresponding to each independent variable)

In mathematics, a function is a binary relation between two sets that associates each element of the first set to exactly one element of the second set.

Since every value of g-¹(x) (the output) corresponds to two values of x (the input),

g-¹(x) is not a function.

Your answer sounds okay to me.

For f(x) , each element in the domain corresponds to exactly one unique element in the codomain.

Every element in the codomain has been mapped to by at least 1 element in the domain (to be precise, it is exactly 1)


This means that it is both injective and surjective.

f(x) is a bijective function.

Okay thank you!