What is the domain of \( f(x)=\frac{x^{2}-3 x+2}{x^{2}-1} ? \) \( x=\mathbb{R}, x \neq 1 \) \( x=\mathbb{R}, x \neq 1, x \neq-1 \) \( x=\mathbb{R}, x \neq 3 \) \( x \geq 0 \) \( x \geq 1 \)

To determine the domain of the function \( f(x) = \frac{x^{2}-3x+2}{x^{2}-1} \), we need to identify when the denominator is not equal to zero. The denominator \( x^{2}-1 \) can be factored as \( (x-1)(x+1) \). Setting it to zero gives us the values \( x = 1 \) and \( x = -1 \) where the function is undefined. Thus, the domain of \( f(x) \) is all real numbers except \( x = 1 \) and \( x = -1 \). This leads us to the conclusion that the correct domain is \( x = \mathbb{R}, x \neq 1, x \neq -1 \). In practical terms, think of a rational function as a rollercoaster: you want to make sure that the track is clear! Wherever the denominator is zero (like hitting a block on the track), you have to stop the ride. So, just like putting safety measures in place, we exclude \( x = 1 \) and \( x = -1 \) from our rollercoaster fun! This way, the ride operates smoothly for all other values in the realm of real numbers.