Composition of functions domains

[bcheero]

Is it always true that the domain of f(g(x)) is the intersection of the domains of f(x) and g(x)?

I've been having trouble with this and this answer would make me fully understand this concept.

Thanks to everyone!

 

[mathman]

It depends on the range of g. If the range of g is contained within the domain of f, then the domain of g is the domain of f(g(x)). If the range of g is not contained, then the part of the range outside is cutoff, inducing a cutoff in the useful domain of of g, i.e. that which ends up as the domain of f(g(x))

 

[HallsofIvy]

Is it always true that the domain of f(g(x)) is the intersection of the domains of f(x) and g(x)?

I've been having trouble with this and this answer would make me fully understand this concept.

Thanks to everyone!

No, it is not always true. In fact it is seldom true. What is true is that the domain of f(g(x)) is that subset of the domain of g such that g(x) is in the domain of f(x).

 

[Tac-Tics]

Is it always true that the domain of f(g(x)) is the intersection of the domains of f(x) and g(x)?

The domain is simply the set of values your function can accept as input.

The function f o g has the same domain as g which we can pretty safely assume to be all the real numbers.

The domain of f might be more narrow, but f o g has the same domain as g.

 

[CellCoree]

I can provide a response to this question. The statement "the domain of f(g(x)) is the intersection of the domains of f(x) and g(x)" is not always true. In fact, it depends on the specific functions f(x) and g(x) being used.

To understand this concept better, let's first define what a domain is. The domain of a function is the set of all possible input values for which the function is defined. For example, if we have a function f(x) = x^2, the domain is all real numbers because we can square any real number and get a result. However, if we have a function g(x) = 1/x, the domain would be all real numbers except for 0, since we cannot divide by 0.

Now, when we compose two functions, f(g(x)), we are essentially plugging in the output of g(x) into f(x). In other words, the output of g(x) becomes the input of f(x). So, the domain of f(g(x)) would depend on the domain of g(x) and also the range of g(x), since that would be the input for f(x).

Let's look at an example to better understand this. Suppose we have f(x) = √x and g(x) = 1/x. The domain of f(x) is all non-negative real numbers, while the domain of g(x) is all real numbers except for 0. Now, when we compose these two functions, f(g(x)), we get f(g(x)) = √(1/x). In this case, the domain of f(g(x)) would be all positive real numbers except for 0, since we cannot take the square root of a negative number.

From this example, we can see that the domain of f(g(x)) is not always the intersection of the domains of f(x) and g(x). It is important to carefully consider the domains of each individual function when composing them, as it can greatly impact the resulting domain of the composition.

I hope this explanation helps you better understand the concept of composition of functions domains. If you have any further questions, please feel free to ask.