4) The range of the function \( f: f(x)=\frac{3 x^{2}-3}{x^{2}-1} \) is ............ \( \begin{array}{llll}\text { (a) } \mathbb{R}-\{1,-1\} & \text { (b) } \mathbb{R}-\{3,-3\} & \text { (c) }\{3,-3\} & \text { d) }\{3\}\end{array} \)

To find the range of the function \( f(x)=\frac{3 x^{2}-3}{x^{2}-1} \), we first simplify it. Notice that we can factor the numerator and denominator: \[ f(x) = \frac{3(x^2 - 1)}{x^2 - 1} = 3 \text{ for } x^2 - 1 \neq 0. \] This means that \( f(x) = 3 \) for all \( x \) except for the points where \( x^2 - 1 = 0 \) (which occur at \( x=1 \) and \( x=-1 \)). At these points, the function is undefined, causing the output to not include the value \( 3 \). Therefore, the range of the function is simply \( \mathbb{R} - \{3\} \). The answer choices you provided seem incorrect regarding the range of the function. It should state that the range is all real numbers except for \( 3 \). So, based on your options, the closest fit seems not to be explicitly listed, but could align with how the other options look if interpreted. To verify the findings, you can try plugging in various \( x \) values into the function and examine the outcomes, which reinforce that \( f(x) \) skims around \( 3 \) yet never touches it.