Question

If f(x) = x2 – 25 and g(x) = x – 5, what is the domain of (StartFraction f Over g EndFraction) (x)?
all real values of x
all real values of x except x = 5
all real values of x except x = –5
all real values of x except x = 5 and x = –5

Answer 1

To find the domain of the function \(\frac{f(x)}{g(x)}\), we need to examine the functions \(f(x)\) and \(g(x)\) provided:

1. \(f(x) = x^2 - 25\)
2. \(g(x) = x - 5\)

The domain of the combined function \(\frac{f(x)}{g(x)}\) will be all real numbers except where the denominator \(g(x)\) is equal to zero, because division by zero is undefined.

Now, let's determine when \(g(x) = 0\):

\[
g(x) = x - 5 = 0
\]

Solving for \(x\):

\[
x = 5
\]

Thus, \(g(x) = 0\) when \(x = 5\). There are no restrictions on the function \(f(x)\), since \(f(x)\) is a quadratic function which is defined for all real numbers.

Since the only restriction in the domain comes from \(g(x)\), the domain of \(\frac{f(x)}{g(x)}\) is all real values of \(x\) except \(x = 5\).

Thus, the correct answer is:

**all real values of x except x = 5**.