If [tex]\( f(x) = x^2 - 25 \)[/tex] and [tex]\( g(x) = x - 5 \)[/tex], what is the domain of [tex]\(\left(\frac{f}{g}\right)(x) ?\)[/tex]

A. all real values of [tex]\( x \)[/tex]

B. all real values of [tex]\( x \)[/tex] except [tex]\( x = 5 \)[/tex]

C. all real values of [tex]\( x \)[/tex] except [tex]\( x = -5 \)[/tex]

D. all real values of [tex]\( x \)[/tex] except [tex]\( x = 5 \)[/tex] and [tex]\( x = -5 \)[/tex]



Answer :

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]