Answer:
D=(-∞,∞)
Step-by-step explanation:
We have the expressions:
[tex]f(x)=x^2-25\\g(x)=x-5[/tex]
and we have to calculate [tex](\frac{f}{g})(x)[/tex] to know the domain.
The domain of a function is all real numbers except where the expression is undefined.
Then,
[tex](\frac{f}{g})(x)=\frac{f(x)}{g(x)}[/tex]
[tex]\frac{f(x)}{g(x)}=\frac{x^2-25}{x-5}[/tex]
Observation: We can use difference of squares in f(x).
Difference of squares is: [tex]a^2-b^2=(a+b)(a-b)[/tex], then
[tex]f(x)=x^2-25\\f(x)=x^2-5^2\\f(x)=(x+5)(x-5)[/tex]
Rewriting the equation:
[tex]\frac{f(x)}{g(x)}=\frac{x^2-25}{x-5}\\\\=\frac{(x+5)(x-5)}{(x-5)}[/tex]
Simplifying:
[tex]\frac{f(x)}{g(x)}=\frac{(x+5)(x-5)}{(x-5)}=x+5[/tex]
We can see that the expression is defined for all real numbers. Then the domain of [tex](\frac{f}{g})(x)[/tex] is all real numbers.
Domain: D=(-∞,∞)
