Find Domain of f o g: Solve x^2+11

[nate808]

I am having problems with finding the domain of the composition of f and g(fog) for example, f(x)=x+7 and g(x)=x^2+4, the fog(x)=x^2+11, but from there i don't know how to find the domain, please help

 

[Hurkyl]

Can you find the domain of f or g individually?

 

[nate808]

ya--in this case both would be all reals, and sow ould this particular example, i guess my main question is if they weren't all reals, like the square root of x and other variations

 

[HallsofIvy]

Since you have g(f(x)), x MUST be in the domain of f in order for f(x) to be defined- that is, the domain of g(f) is a subset of the domain of f.

Then, however, f(x) must be such that g(f(x)) is defined. Start with the domain of f and delete those values for which g(f(x)) is not defined. Exactly how you do that depends on the specific values of f and g.

In your original example, the domains of both g and f are 'all real numbers' so the domain of g(f) is also 'all real numbers'.

If you had g(x)= sqrt(x), f(x)= 3x- 1, the domain of f is all real numbers but the domain of g is only non-negative numbers. For what x is 3x-1>= 0?

Conversely, if g(x)= 3x-1, f(x)= sqrt(x), The domain of f is non-negative numbers. Since g(x) is defined for all x, g(f(x)) is defined for all numbers for which f is defined: non-negative numbers.