In the construction of $\operatorname{Frac}(R)$, where $R$ is a domain, we define a partition on $R \times R^\times$ where $R^\times:= R \setminus \{0\}$. which in turn becomes a field containing $R$.
Taking $R:=\mathbb{Q}[x]$, can we tell that $\frac{x^2+1}{x-1} \in \mathbb{Q}(x)$? As $x-1$ is not the zero-polynomial it is a member of $\mathbb{Q}[x]^\times$, hence $\frac{x^2+1}{x-1}$ should be a member of $\mathbb{Q}(x)$. But this is undefined when 1 is plugged in $x$.
Where am I going wrong?
On
Your understanding has a gap starting at the ring of polynomials $\mathbb Q[X]$, not at the field of fractions. So I'm going to focus on the former, not the latter.
To start with, polynomials are not functions. Memorize this sentence and never forget it: Polynomials are not functions.
What polynomials really are should be understood through first understanding what their purpose is. A polynomial's purpose is to be a template for expressions that involve some element $X$ of a ring using only the operations provided by the ring: addition, subtraction and multiplication. We are allowed to multiply $X$ by an element of the ring or by itself, and we can add or subtract the results of these multiplications. The end result is a polynomial expression in $X$ with coefficients in the underlying ring. This expression is a template into which we want to plug in any object for which the ring operations addition, subtraction and multiplication are meaningful. For instance, real matrices can be multiplied by themselves or by real numbers, and they can be added. So matrices are a kind of object we might want to plug into a polynomial. But then the result is a matrix, so it wouldn't make the least bit of sense to define the polynomial to be a function mapping numbers to numbers. On the other hand, we might as well plug in numbers, and then the results are numbers. So we also can't define polynomials to be functions mapping matrices to matrices. Rather, polynomials should be something that allows us to plug in a variety of objects before we even know what kind of object we want to plug in. What kind of map we get shouldn't be predefined.
The takeaway is that polynomials do not serve as maps, but as templates for maps involving all kinds of objects, where we're free to adjust the domain to our liking: we could make it a matrix space, or the ring itself, or something entirely different. The same goes for the fraction field. Its elements are templates for functions whose domain we can adjust dynamically. Simply don't use an expression like $\frac{1}{X}$ as a template for functions whose domain contains $0$ and you're good.
The simple answer is that elements of $\mathbb{Q}(x)$ are formal rational functions. strictly speaking, the values they take on at certain points are irrelevant to them. Their evaluation maps need not be defined for all elements in the underlying field. Another example is that $\frac{1}{x}$ which is the multiplicative inverse of $x$ is not defined as a function at $x=0$. That's fine because the way we are looking at it, $\frac{1}{x}$ is not actually a function, it's really just the formal inverse of $x$.
There is a similar intuition here to how $x^2+x+1\in\mathbb{F}_2[x]$ is not the same thing as $1\in\mathbb{F}_2[x]$ even though the evaluation maps are the same.
$x$ is just something that is transcendental over $\mathbb{Q}$ we think of the elements of $\mathbb{Q}(x)$ as being rational functions, but really we shouldn't be considering any of their properties as functions in the context of algebra. It's helpful to realise that $\mathbb{Q}(x)$ is isomorphic to $\mathbb{Q}(\pi)$