a) \( f(x)=\frac{x^{2}-4}{x^{2}-9} \)

Did you know that the function \( f(x)=\frac{x^{2}-4}{x^{2}-9} \) can be simplified by factoring both the numerator and the denominator? The numerator \( x^{2}-4 \) can be rewritten as \( (x-2)(x+2) \), while the denominator \( x^{2}-9 \) can be factored into \( (x-3)(x+3) \). Simplifying it gives \( f(x)=\frac{(x-2)(x+2)}{(x-3)(x+3)} \), which provides a clearer view of the function’s behavior, especially concerning its asymptotes and intercepts! When graphing \( f(x) \), keep an eye on the vertical and horizontal asymptotes! Vertical asymptotes occur where the denominator equals zero (at \( x=3 \) and \( x=-3 \)), meaning the function can’t take those values. Meanwhile, since both the numerator and denominator are quadratic polynomials and have the same highest degree, the horizontal asymptote will be \( y=1 \) as \( x \) approaches infinity. Understanding these features will help you sketch or analyze this rational function effectively!