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]<br /> <br /> (i) Find the natural domains of $$f, g, f + g, \frac{f}{g}, and \frac{g}{f}$$<br /> <br /> <h2>Homework Equations</h2><br /> <br /> N/A<br /> <br /> <h2>The Attempt at a Solution</h2><br /> <br /> 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.<br /> <br /> 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?<br /> <br /> Thanks in advance,<br /> Charismaztex[/itex]

 

[HallsofIvy]

Homework Statement

3. (a) Let $f(x) = ln(x^2-1)$, and [itex]g(x)=\frac{x}{\sqrt{2-x}}[/tex]<br /> <br /> (i) Find the natural domains of $$f, g, f + g, \frac{f}{g}, and \frac{g}{f}$$<br /> <br /> <h2>Homework Equations</h2><br /> <br /> N/A<br /> <br /> <h2>The Attempt at a Solution</h2><br /> <br /> 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.[/itex]

$x cannot be <b>greater than</b> or equal to 2.<br /> <br /> In order that we be able to add, subtract, or multiply f(x) and g(x), both must be defined. That means that the domain of f+g, f- g or fg is the <b>intersection</b> of their separate domains. In order that we bae able to divide f by g, we must also exclude points where g(x)= 0 and vice versa for g/f. <br /> <br /> <blockquote data-attributes="" data-quote="" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> My question is: what are the rules for manipulation of the domains? Do we simply combine the natural domains, and what about the quotient of 2 natural domains?<br /> <br /> Thanks in advance,<br /> Charismaztex </div> </div> </blockquote>$

 

[Charismaztex]

Thanks :)