To determine the domain of the function [tex]\(\left(\frac{f}{g}\right)(x)\)[/tex], we need to understand where the denominator [tex]\(g(x)\)[/tex] is equal to zero, since division by zero is undefined.
1. Identify the functions:
- [tex]\(f(x) = x^2 - 25\)[/tex]
- [tex]\(g(x) = x - 5\)[/tex]
2. Set the denominator [tex]\(g(x)\)[/tex] to zero and solve for [tex]\(x\)[/tex]:
[tex]\[
g(x) = x - 5 \\
x - 5 = 0 \\
x = 5
\][/tex]
Therefore, [tex]\(g(x) = 0\)[/tex] when [tex]\(x = 5\)[/tex].
3. Determine the domain:
- The domain of [tex]\(\left( \frac{f}{g} \right)(x)\)[/tex] includes all real values of [tex]\(x\)[/tex] except where [tex]\(g(x) = 0\)[/tex], which happens at [tex]\(x = 5\)[/tex].
- As [tex]\(g(5) = 0\)[/tex], [tex]\(\left(\frac{f}{g}\right)(5)\)[/tex] is undefined, and thus [tex]\(x = 5\)[/tex] must be excluded from the domain.
4. Conclusion:
The domain of [tex]\(\left(\frac{f}{g}\right)(x)\)[/tex] is all real values of [tex]\(x\)[/tex] except [tex]\(x = 5\)[/tex].
Hence, the correct domain is:
- all real values of [tex]\(x\)[/tex] except [tex]\(x = 5\)[/tex]
