Domain of f and g?

Question

How do I find the domain of f and g. I have a problem that says Suppose f(x)=x²-5 and g(x)=the square root of x+5. I first have to find (f/g)(4) which I think i just plug 4 in and divide f by g. Then it asks for the domain of f/g which I am confused about. After that it tells me to find (fog)(x) where I am pretty sure I just plug in f in g(x) function equaling x. Then it again asked me for the domain of f o g and I am still unaware of how to find this domain.

Answer 1

OK.  So f(x) = x²-5, and g(x) = √(x+5).

 

f(4) = 16 - 5 = 11, and g(4) = √9 = 3.

 

So (f/g)(4) = 11/3.

 

Since the only restriction on x on the function (f/g)(x) occurs when g(x) ≠0, the domain of this function is x>-5 (since the radical must be non-negative,and the denominator must not be 0).

 

(fog)(x) is the same as (f(g(x)), which means replace the g function in place of the value of x, in f(x).

 

So, (fog)(x) = (f(√(x+5)) = (√(x+5))² - 5 = x+5-5 = x.  Here the domain must be x≥-5, since the domain of g(x) must be in the f(x) and (x+5) cannot be negative.

 

I hope this helps.

 

Answer 2

Hello,

you are correct with (f/g)(4):

(f/g)(4) = f(4)/g(4), so plug in 4 to each function and then divide the result.

 

For the domain of (f/g), you want to find any values for x that does not result in a real number for the function f(x)/g(x) = (x²-5)/(√x+5).  For this example, any x which makes x+5<0 will give an imaginary number.  Additionally x=-5 will give an undefined result.  And so the domain is (-5,∞), or x>-5.

 

again you are correct that (fοg)(x) is f(g(x)) or just x once you plug g(x) into f(x).  Thus the domain for (fog)(x) is x>-5 since the domain of each is carried through