Exercise 1.1.2

Since the domains of the functions $f$ and $g$ are not specified explicitly, we assume that the domains of both functions are their maximal intervals of existence. Let’s calculate them.

• Because the denominator can not be equal to 0, for $f$, we have

  $$\forall <!--\; FUNCTION\; APPLICATION\; -->x\in ℝ:x-1\ne 0⟺x\ne 1.$$

  In other words, the domain interval of the function $f$ is

  $$\left\{x\in ℝ\mid x\ne 1\right\}=(-\infty ,1)\cup (1,+\infty ).$$

• The domain of the function $g$ is $(-\infty ,+\infty ).$

Since functions, by definition, can be equal only if they have the same domain, we have $f\ne g.$