Which of the following functions is an invert | vedclass

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(A) function is invertible if and only if it is a bijection (both one-to-one and onto).$1$. For $f(x) = \frac{1}{x - 1}$,the domain is $x \in \mathbb{

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$f(x) = \frac{1}{x - 1}$

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(A) function is invertible if and only if it is a bijection (both one-to-one and onto).$1$. For $f(x) = \frac{1}{x - 1}$,the domain is $x \in \mathbb{R} \setminus \{1\}$ and the range is $y \in \mathbb{R} \setminus \{0\}$. This function is one-to-one and onto,hence it is invertible.$2$. For $f(x) = x^2$ defined for all $x \in \mathbb{R}$,it is not one-to-one because $f(1) = f(-1) = 1$. Thus,it is not invertible.$3$. For $f(x) = x^2$ with $x \ge 0$ or $x \le 0$,the function becomes a bijection on its restricted domain and codomain,but the standard definition of an invertible function usually refers to the function being bijective over its natural domain. However,among the given choices,$f(x) = \frac{1}{x - 1}$ is the most standard example of a function that is bijective on its entire domain.

Which of the following functions is an invertible function?

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