Question

If f(x) = x^2 – 25 and g(x) = x – 5, what is the domain of (f/g)(x)

Answer 1

(f/g)(x) = (x^2-25)/(x-5) = x+5. So domain will be R, all real numbers

Answer 2

D=(-∞,∞)

Explanation:

We have the expressions:

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

and we have to calculate
`((f)/(g))(x)` to know the domain.

The domain of a function is all real numbers except where the expression is undefined.

Then,

`((f)/(g))(x)=(f(x))/(g(x))`

`(f(x))/(g(x))=(x^2-25)/(x-5)`

Observation: We can use difference of squares in f(x).

Difference of squares is:
`a^2-b^2=(a+b)(a-b)`, then

`f(x)=x^2-25\\f(x)=x^2-5^2\\f(x)=(x+5)(x-5)`

Rewriting the equation:

`(f(x))/(g(x))=(x^2-25)/(x-5)\\\\=((x+5)(x-5))/((x-5))`

Simplifying:

`(f(x))/(g(x))=((x+5)(x-5))/((x-5))=x+5`

We can see that the expression is defined for all real numbers. Then the domain of
`((f)/(g))(x)` is all real numbers.

Domain: D=(-∞,∞)