## Introduction

In mathematics, the composition of two functions is the result of applying one function after the other. If f is a function from A to B and g is a function from B to C, then the composition of f and g, denoted by g ∘ f (read as “g of f”), is the function from A to C such that (g ∘ f)(x) = g(f(x)) for all x in A.

## What is the domain of a function?

The domain of a function is the set of all input values for which the function produces a result. In other words, it is the set of all values that can be plugged into a function in order to produce a meaningful result. For example, the domain of the function f(x) = x2 is all real numbers, because no matter what value you plug into this function, it will always give you a real number result. On the other hand, the domain of the function g(x) = 1/x is all real numbers except for 0, because if you plug 0 into this function, you will get an undefined result (i.e., division by 0 is not defined). Therefore, the domain of the function g(x) = 1/x is all real numbers except for 0.

## How to find the domain of a function?

To find the domain of a function, we need to find all the values of x for which the function produces a real result. We can do this by looking at the function and seeing what values of x will make the function undefined.

In this case, we can see that if x is less than -4 or if x is greater than 4, then the function will be undefined. Therefore, the domain of this function is:

{x | -4 < x < 4}

## How to find the domain of a function when it is not given?

There are a few ways that you can find the domain of a function when it is not given. One way is to simply look at the function and try to determine what inputs would make the function undefined. Another way is to use algebra to solve for the domain.

For example, let’s say we have the function f(x) = x^2 + 16 and we want to find the domain of this function. We can see that this function would be undefined if we tried to input a negative number into it, since we would be taking the square root of a negative number. Therefore, our domain is all real numbers except for negative numbers. We could also write this as {x | x >= 0}.

Similarly, if we have the function g(x) = x^4 + 4, we can see that this function would be undefined if we tried to input a negative number into it. Therefore, our domain is all real numbers except for negative numbers. We could also write this as {x | x >= 0}.

Now that we know the domains of both functions, we can find the domain of their composition, fg(x). The domain of fg(x) will be all real numbers except for those values of x where either f(x) or g(x) is undefined. In other words, the domain of fg(x) will be {x | x >= 0}.

## What is the domain of a composition of functions?

The domain of a composition of functions is the set of input values for which both functions in the composition are defined. For example, if f(x) = x2 and g(x) = x4, then the domain of the composition fg would be the set of input values for which both x2 and x4 are defined. This would typically be the set of real numbers excluding any values where either function is undefined (such as negative numbers for square roots).

## How to find the domain of a composition of functions?

To find the domain of a composition of functions, you need to find the intersection of the domains of the individual functions. In other words, the domain of the composition is the set of all values for which both functions are defined.

For example, let’s say you have two functions: f(x) = x2 + 16 and g(x) = x4. The domain of f(x) is all real numbers, and the domain of g(x) is all real numbers except for when x = 0 (because you can’t take the square root of a negative number).

Therefore, the domain of the composition fg(x) is all real numbers except for when x = 0.

## How to find the domain of a composition of functions when it is not given?

To find the domain of a composition of functions f ∘ g, where the domain of g is not given, we must first find the domain of g. To do this, we need to determine what values of x will make g(x) well-defined. In other words, we need to find the values of x that will result in a real value for g(x).

For this example, let’s say that f(x) = x2 + 16 and g(x) = 4x. We can see that the domain of f is all real numbers since we can square any real number and add 16 to it and get a real result. However, for g(x), we need to be careful. We can’t just take any value for x and multiply it by 4 because we could end up with an imaginary number. So, in order to have a real result for g(x), x must be such that 4x is also a real number. This means that the domain of g is all real numbers except for those values of x that would make 4x an imaginary number. In other words, the only values of x that are not in the domain of g are those where 4x is not a real number.

Now that we know the domain of both f and g, we can find the domain of their composition by taking the intersection of their domains. In this case, both functions have the same domain (all real numbers), so their composition also has this domain.

## Conclusion

In conclusion, to find fg, you must first find the domain of each function. The domain of f is all real numbers except for when x=4, and the domain of g is all real numbers except for when x=-2. Therefore, the domain of fg is all real numbers except when x=4 or x=-2.