A = {1, 2, 3} which of the following function f: A → A does not have an inverse function

The correct option is {"{(1, 2), (2, 1), (3, 1)}"}.

To understand why this function does not have an inverse, we need to look at the mathematical definition of an invertible function.

A function $f:A\to A$ has an inverse function if and only if it is a bijective function. A bijective function must satisfy two conditions:
1. Injective (One-to-One): Each element in the domain $A$ maps to a unique element in the codomain $A$. That is, if $x\ne y$, then $f\left(x\right)\ne f\left(y\right)$.
2. Surjective (Onto): Every element in the codomain $A$ has at least one pre-image in the domain $A$.

Let's analyze the given function $f=\left\{\right(1,2),(2,1),(3,1\left)\right\}$:
- Element $1$ maps to $2$ (i.e., $f\left(1\right)=2$).
- Element $2$ maps to $1$ (i.e., $f\left(2\right)=1$).
- Element $3$ maps to $1$ (i.e., $f\left(3\right)=1$).

From this mapping, we can see that:
$$f\left(2\right)=f\left(3\right)=1$$
Since two distinct input values ($2$ and $3$) map to the same output value ($1$), the function is not injective (not one-to-one).

Furthermore, looking at the codomain $A=\{1,2,3\}$, the element $3$ is never mapped to by any element in the domain. Therefore, the function is also not surjective (not onto).

Because the function is neither one-to-one nor onto, it is not bijective, and consequently, it does not have an inverse function.