#### Question

Let *f*: *X* → *Y* be an invertible function. Show that* f *has unique inverse. (Hint: suppose *g*_{1} and *g*_{2} are two inverses of *f*. Then for all *y* ∈ *Y*, *f*o*g*_{1}(*y*) = I_{Y}(*y*) = *f*o*g*_{2}(*y*). Use one-one ness of *f*).

#### Solution

Let *f*: *X* → *Y* be an invertible function.

Also, suppose *f* has two inverses (say `g_1` and `g_2)`_{).}

Then, for all *y* ∈*Y*, we have:

`fog_1(y) = I_y (y) = fog_2 (y)`

`=> f(g_1(y)) = f(g_2(y))` [f is invertible => f is one-one]

`=> g_1 = g_2` [g is one- one]

Hence,* f* has a unique inverse.

Is there an error in this question or solution?

Solution Let F: X → Y Be an Invertible Function. Show that F Has Unique Inverse. (Hint: Suppose G1 and G2 Are Two Inverses of F. Then for All Y ∈ Y, Fog1(Y) = Iy(Y) = Fog2(Y). Use One-one Ness of F). Concept: Composition of Functions and Invertible Function.