How do you find the domain of function operations?

[xnonamex0206]

Homework Statement

Find a. (f+g)(x) b. (f-g)(x) c. (fg)(x) d. (f/g)(x)

f(x)=2x-5 & g(x)=4

Homework Equations

(f+g)(x)=2x-1
(f-g)(x)=2x-9
(fg)(x)=8x-20
(f/g)(x)=2x-5/4

The Attempt at a Solution

When I've completed all the other equations, I'm stuck on domain. I'm thinking that its all reals, but I'm not quite sure on how this is true. Is it because it cannot equal zero, or a non-negative number?

In other other words how exactly do you find the domain?

 

[slider142]

Just a little note, they are actually asking for the *largest* possible domain of each function over the reals, as you can give a function any domain you want, as long as the function is defined for every element of its domain.
Is there any value of x for which any of your functions does not yield a real number? If they exist, these x's cannot be in any domain of your function.
Hint: The identity function I(x) = x is defined for all real numbers x. Your functions are just real numbers added to real multiples of this function. Ie., it boils down to showing that if x is a real number, then kx is also a real number for real k.
PS. Your (f/g)(x) is missing the division by 4.

 

[Almanzo]

fg, f+g and f-g are defined everywhere, where f and g are both defined. f/g is defined everywhere, where f and g are both defined and g does not equal zero.

 

[xnonamex0206]

Just a little note, they are actually asking for the *largest* possible domain of each function over the reals, as you can give a function any domain you want, as long as the function is defined for every element of its domain.
Is there any value of x for which any of your functions does not yield a real number? If they exist, these x's cannot be in any domain of your function.
Hint: The identity function I(x) = x is defined for all real numbers x. Your functions are just real numbers added to real multiples of this function. Ie., it boils down to showing that if x is a real number, then kx is also a real number for real k.
PS. Your (f/g)(x) is missing the division by 4.

What does the k represent? So if x is a real number then the domain should include all reals? So basically to find the domain you should plug in some x's to see if they aren't real numbers?

I fixed the (f/g) forgot to add the /4.

fg, f+g and f-g are defined everywhere, where f and g are both defined. f/g is defined everywhere, where f and g are both defined and g does not equal zero.

What!?

 

[HallsofIvy]

Yes, xnoname0206 is correct. If the domains of f and g are not the same, the domains of fg, f+ g and f-g are the intersections of the domains of f and g. That is what xnoname0206 meant by "where f and g are both defined". f/g is defined on the intersection of the domains minus points where g(x)= 0.