Question

How do you verify if `f(x)=3-x; g(x)=3-x` are inverse functions?

Answer 1

Please see below.

Explanation:

This is a very interesting case. Both the equations `f(x)=3-x` and `g(x)=3-x` can be written as `x+y=3`, where `y` replaces `f(x)` or `g(x)`.

Observe that in `x+y=3`, if `x` is changed to `y` and vice-versa, the equation does not change and hence such equations are inverses of each other.

These are also symmetric w.r.t. `y=x`, as may be seen from the following.

graph{(x+y-3)(x-y)=0 [-9.5, 10.5, -3.28, 6.72]}

Let us have another such function like `x^2+xy+y^2=10`. Observe that in this too as if `x` is changed to `y` and vice-versa, the equation does not change. This may be seen from the graph as well.

graph{(x^2+xy+y^2-10)(x-y)=0 [-9.54, 10.46, -5.64, 4.36]}