For the given rational function \( f(x)=\frac{12-4 x^{2}}{x^{2}-4} \) (A) Find any intercepts for the graph. (B) Find any vertical and horizontal asymptotes for the graph. (C) Graph the function. A. The \( x \)-intercept(s) is(are) \( x=\square \). (Type an exact answer. Use a comma to separate answers as needed.) B. There are no \( x \)-intercepts. Identify the \( y \)-intercepts, if there are any. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The \( y \)-intercept(s) is(are) \( y=\square \) (Type an exact answer. Use a comma to separate answers as needed.) B. There are no \( y \)-intercepts.

To find the \( x \)-intercepts of the function \( f(x)=\frac{12-4 x^{2}}{x^{2}-4} \), we set the numerator equal to zero: \[ 12 - 4x^2 = 0 \implies 4x^2 = 12 \implies x^2 = 3 \implies x = \pm\sqrt{3}. \] So the \( x \)-intercepts are \( x=\sqrt{3}, -\sqrt{3} \). Next, for the \( y \)-intercept, we evaluate \( f(0) \): \[ f(0) = \frac{12 - 4(0)^2}{(0)^2 - 4} = \frac{12}{-4} = -3. \] Thus, the \( y \)-intercept is \( y=-3 \). To summarize: A. The \( x \)-intercept(s) is(are) \( x=\sqrt{3}, -\sqrt{3} \). A. The \( y \)-intercept(s) is(are) \( y=-3 \). Now, let's talk about some interesting background and common mistakes when working with rational functions! Rational functions have been a staple of mathematical study for centuries and originate from the exploration of ratios of polynomials. The structure of these functions can often lead to fascinating behavior, such as asymptotes and discontinuities. Understanding these characteristics is crucial for mastering more advanced mathematical concepts. A common mistake when finding intercepts is overlooking the need to check each component of the function. Students often forget to simplify the function before searching for intercepts or asymptotes, which can lead to incorrect conclusions about where the graph intersects the axes or how it behaves at infinity. Remember to factor completely and cancel before evaluating!