Identify any real numbers excluded from the domain of the rational function. (Enter your answers as a comma-separated list. If no real numbers are excluded, enter NONE.) \[ f(x)=\frac{x-6}{(x-7)(x+3)} \] \( x=\square \) Additional Materials

To find the excluded values from the domain of the rational function \( f(x)=\frac{x-6}{(x-7)(x+3)} \), we need to identify when the denominator is equal to zero, as division by zero is undefined. The denominator is \( (x-7)(x+3) \). Setting each factor to zero, we find: 1. \( x - 7 = 0 \) ⇒ \( x = 7 \) 2. \( x + 3 = 0 \) ⇒ \( x = -3 \) Thus, the real numbers excluded from the domain are \( x = 7, -3 \). Here’s a neat way to think about rational functions: they’re like party hosts who can only invite people to their event if they don’t have a conflict! In this case, the function can’t handle the values where the denominator equals zero, causing a disruption at \( x = 7 \) and \( x = -3 \). Now, if you ever encounter similar functions, always remember: just look for those troublesome zeroes in the denominator! It’s a shortcut that helps you avoid any awkward encounters in the math world.