Answer :
To determine the domain of [tex]\(\left(\frac{f}{g}\right)(x)\)[/tex] where [tex]\(f(x) = x^2 - 25\)[/tex] and [tex]\(g(x) = x - 5\)[/tex], we need to understand when the function [tex]\(\left(\frac{f}{g}\right)(x)\)[/tex] is defined.
A rational function [tex]\(\left(\frac{f}{g}\right)(x)\)[/tex] is defined whenever the denominator [tex]\(g(x)\)[/tex] is not zero. So we first need to identify the values of [tex]\(x\)[/tex] that make [tex]\(g(x) = 0\)[/tex].
Given:
[tex]\[ g(x) = x - 5 \][/tex]
We solve for [tex]\(x\)[/tex] when [tex]\(g(x) = 0\)[/tex]:
[tex]\[ x - 5 = 0 \][/tex]
[tex]\[ x = 5 \][/tex]
Thus, [tex]\(g(x) = 0\)[/tex] when [tex]\(x = 5\)[/tex]. This means that [tex]\(\left(\frac{f}{g}\right)(x)\)[/tex] is undefined when [tex]\(x = 5\)[/tex].
Next, we check if there are any other restrictions on [tex]\(x\)[/tex] that could make the function [tex]\(\left(\frac{f}{g}\right)(x)\)[/tex] undefined. Specifically, we consider any values of [tex]\(x\)[/tex] that might make [tex]\(f(x)\)[/tex] or [tex]\(g(x)\)[/tex] undefined. However, since both [tex]\(f(x) = x^2 - 25\)[/tex] and [tex]\(g(x) = x - 5\)[/tex] are polynomials, they are defined for all real [tex]\(x\)[/tex].
Consequently, the only restriction on the domain of [tex]\(\left(\frac{f}{g}\right)(x)\)[/tex] comes from the denominator being zero, which happens when [tex]\(x = 5\)[/tex].
Therefore, the domain of [tex]\(\left(\frac{f}{g}\right)(x)\)[/tex] is:
[tex]\[ \text{all real values of } x \text{ except } x=5 \][/tex]
So the correct answer is:
[tex]\[ \boxed{\text{all real values of } x \text{ except } x=5} \][/tex]
A rational function [tex]\(\left(\frac{f}{g}\right)(x)\)[/tex] is defined whenever the denominator [tex]\(g(x)\)[/tex] is not zero. So we first need to identify the values of [tex]\(x\)[/tex] that make [tex]\(g(x) = 0\)[/tex].
Given:
[tex]\[ g(x) = x - 5 \][/tex]
We solve for [tex]\(x\)[/tex] when [tex]\(g(x) = 0\)[/tex]:
[tex]\[ x - 5 = 0 \][/tex]
[tex]\[ x = 5 \][/tex]
Thus, [tex]\(g(x) = 0\)[/tex] when [tex]\(x = 5\)[/tex]. This means that [tex]\(\left(\frac{f}{g}\right)(x)\)[/tex] is undefined when [tex]\(x = 5\)[/tex].
Next, we check if there are any other restrictions on [tex]\(x\)[/tex] that could make the function [tex]\(\left(\frac{f}{g}\right)(x)\)[/tex] undefined. Specifically, we consider any values of [tex]\(x\)[/tex] that might make [tex]\(f(x)\)[/tex] or [tex]\(g(x)\)[/tex] undefined. However, since both [tex]\(f(x) = x^2 - 25\)[/tex] and [tex]\(g(x) = x - 5\)[/tex] are polynomials, they are defined for all real [tex]\(x\)[/tex].
Consequently, the only restriction on the domain of [tex]\(\left(\frac{f}{g}\right)(x)\)[/tex] comes from the denominator being zero, which happens when [tex]\(x = 5\)[/tex].
Therefore, the domain of [tex]\(\left(\frac{f}{g}\right)(x)\)[/tex] is:
[tex]\[ \text{all real values of } x \text{ except } x=5 \][/tex]
So the correct answer is:
[tex]\[ \boxed{\text{all real values of } x \text{ except } x=5} \][/tex]
