Finding the sum and quotient of 2 natural domains

[Charismaztex]

Homework Statement

3. (a) Let $f(x) = ln(x^2-1)$, and [itex]g(x)=\frac{x}{\sqrt{2-x}}[/tex]

(i) Find the natural domains of $$f, g, f + g, \frac{f}{g}, and \frac{g}{f}$$

Homework Equations

N/A

The Attempt at a Solution

I know that the natural domain of f(x) is x belongs to real numbers and that x greater than 1 and less than -1 (ln(0) and ln(-ve no.) is undefined). The natural domain of g(x) is that x belongs real numbers and that x cannot be 2.

My question is: what are the rules for manipulation of the domains? Do we simply combine the natural domains, and what about the quotient of 2 natural domains?

Thanks in advance,
Charismaztex

 

[HallsofIvy]

Homework Statement

3. (a) Let $f(x) = ln(x^2-1)$, and [itex]g(x)=\frac{x}{\sqrt{2-x}}[/tex]

(i) Find the natural domains of $$f, g, f + g, \frac{f}{g}, and \frac{g}{f}$$

Homework Equations

N/A

The Attempt at a Solution

I know that the natural domain of f(x) is x belongs to real numbers and that x greater than 1 and less than -1 (ln(0) and ln(-ve no.) is undefined). The natural domain of g(x) is that x belongs real numbers and that x cannot be 2.

x cannot be greater than or equal to 2.

In order that we be able to add, subtract, or multiply f(x) and g(x), both must be defined. That means that the domain of f+g, f- g or fg is the intersection of their separate domains. In order that we bae able to divide f by g, we must also exclude points where g(x)= 0 and vice versa for g/f.

My question is: what are the rules for manipulation of the domains? Do we simply combine the natural domains, and what about the quotient of 2 natural domains?

Thanks in advance,
Charismaztex

 

[Charismaztex]

Thanks :)