To find the domain of the function
[tex]\[ f(x) = \frac{x+6}{(x-7)(x+5)}, \][/tex]
we need to determine the values of [tex]\( x \)[/tex] for which the function is undefined. This typically occurs where the denominator equals zero, as division by zero is undefined in mathematics.
Here's the step-by-step procedure to find the domain:
1. Identify the Denominator:
The denominator of the function is [tex]\((x-7)(x+5)\)[/tex].
2. Set the Denominator to Zero:
To find the values that make the denominator zero, we solve the equation:
[tex]\[ (x-7)(x+5) = 0. \][/tex]
3. Solve for [tex]\( x \)[/tex]:
We can set each factor equal to zero and solve for [tex]\( x \)[/tex]:
- For the first factor:
[tex]\[ x - 7 = 0 \implies x = 7. \][/tex]
- For the second factor:
[tex]\[ x + 5 = 0 \implies x = -5. \][/tex]
4. Determine the Domain:
The values [tex]\( x = 7 \)[/tex] and [tex]\( x = -5 \)[/tex] make the denominator zero, thus the function [tex]\( f(x) \)[/tex] is undefined at these points. Therefore, the domain of the function includes all real numbers except [tex]\( x = 7 \)[/tex] and [tex]\( x = -5 \)[/tex].
Hence, the domain of the function [tex]\( f(x) = \frac{x+6}{(x-7)(x+5)} \)[/tex] is:
[tex]\[ \boxed{\text{all real numbers except } -5 \text{ and } 7}. \][/tex]
