Question

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

Explanation:

Step1: Identify undefined points

A rational function is undefined when its denominator is zero. So, set the denominator \((x - 7)(x + 5)\) equal to zero.
\[
(x - 7)(x + 5)=0
\]

Step2: Solve for x

Using the zero - product property (if \(ab = 0\), then either \(a = 0\) or \(b = 0\)), we have two equations:

• Case 1: \(x-7 = 0\), which gives \(x = 7\).
• Case 2: \(x + 5=0\), which gives \(x=-5\).

The function is undefined at \(x = 7\) and \(x=-5\). So the domain of the function is all real numbers except \(x=-5\) and \(x = 7\).

Answer:

C. all real numbers except \(-5\) and \(7\)