Question

What is the domain of the function?
f(x) = x+6 / (x -7)(x+5)
all real numbers except 7
all real numbers except 5 and - 7
all real numbers except -5 and 7
all real numbers

Answer 1

To determine the domain of the function

\[
f(x) = \frac{x + 6}{(x - 7)(x + 5)},
\]

we need to identify the values of \(x\) that make the denominator zero, as these values will not be included in the domain.

The denominator is

\[
(x - 7)(x + 5).
\]

Setting the denominator equal to zero:

\[
(x - 7)(x + 5) = 0.
\]

This gives us two equations to solve:

1. \(x - 7 = 0 \implies x = 7\)
2. \(x + 5 = 0 \implies x = -5\)

Therefore, the function is undefined at \(x = 7\) and \(x = -5\).

The domain of the function \(f(x)\) consists of all real numbers except those values that make the denominator zero. Thus, the domain is:

**All real numbers except -5 and 7.**

So the correct choice would be "all real numbers except -5 and 7".