Find the domain of the function \( f(x)=\frac{x+1}{(x-3)(x+4)} \) a) \( (-\infty,-4) \cup(-4,-1) \cup(-1,3) \cup(3, \infty) \) b) \( (-\infty,-4) \cup(3, \infty) \) c) \( (-\infty,-3) \cup(-3,4) \cup(4, \infty) \) d) \( (-\infty,-4) \cup(-4,3) \cup(3, \infty) \)

To find the domain of the function \( f(x) = \frac{x+1}{(x-3)(x+4)} \), we need to identify the values of \( x \) that make the denominator zero, as these values cannot be included in the domain. The denominator is zero when: 1. \( x - 3 = 0 \) → \( x = 3 \) 2. \( x + 4 = 0 \) → \( x = -4 \) Thus, \( x = 3 \) and \( x = -4 \) are the values that must be excluded from the domain. Therefore, the domain is all real numbers except \( -4 \) and \( 3 \). So, the domain of the function is: **Answer: a) \( (-\infty,-4) \cup(-4,-1) \cup(-1,3) \cup(3, \infty) \)** But since it doesn't include -4 and 3, the correct answer is actually: **Answer: d) \( (-\infty,-4) \cup(-4,3) \cup(3, \infty) \)** --- Back in time, rational functions like this one became essential in mathematics as they allowed for the modeling of real-world relationships, leading to advances in fields such as physics and economics! Remember, knowing the domain helps you avoid those pesky undefined points. Using rational functions can be practical in everyday scenarios, like dispensing medication where dosages depend on specific values or calculating rates, where you must ensure that certain conditions (like avoiding division by zero) are met for reliable results. Understanding domain highlights the importance of scenarios we can reasonably apply formulas to!