Domain of composition

Question

If the domain of g(x) is the set Dg={x|x|}≠0 and the domain of f(x) is the entire number line, then

 

a. what is the domain of the composition f(g(x))?

 

b. what condition defines the domain of g(f(x))?

Answer 1

Hi, Naomi.

 

a. f(g(x)) means that we are plugging the function g(x) into f(x). Whatever values that would make g(x) undefined would also make f(x) undefined. So, in this case, our domain is {x|x doesn't equal 0}.

 

b. On the reverse, g(f(x)) means we are plugging f(x) into g(x). The range of f(x) takes the role of the x in g(x). So, our domain is {f(x)|f(x) doesn't equal 0}. Any values that would make f(x) = 0 would make g(x) undefined.

 

For example, if f(x) = x - 5, the value 5 would make f(x) = 0 and therefore, g(f(5)) would be undefined.

 

 

Let me know if this is all clear.