What is the domain of the function? \[ f(x)=\frac{x+6}{(x-7)(x+5)} \] A all real numbers except 7 B all real numbers except 5 and -7 C all real numbers except -5 and 7 D all real numbers

To find the domain of the function \( f(x)=\frac{x+6}{(x-7)(x+5)} \), we need to identify the values of \( x \) that would make the denominator zero, as division by zero is undefined. The denominator \((x-7)(x+5)\) becomes zero when \( x = 7 \) or \( x = -5\). Therefore, the domain includes all real numbers except these values. The correct answer is C, all real numbers except -5 and 7. Now that we’ve tackled the domain, let’s make it more fun! When determining the domain of functions like \( f(x) \), think of it as setting up a party: you're inviting all the numbers except for those unruly guests (where the denominator equals zero) who would cause a scene! Just kick them out, and you're good to go! In practical terms, understanding the domain can prevent headaches in real-world scenarios such as physics or engineering problems where certain calculations become impossible if you try to plug in forbidden values. Avoid those pesky pits of doom by knowing your domain well!