What values should be excluded from the domain of composite functions?

Answer 1

If we consider the composition of two functions f and g, the domain of y = f (g(x)) consists of all x which are in the domain of g such g(x) is also in the domain of f.

If both functions have a domain of all real numbers, this problem is fairly simple.

For example if f and g are two polynomials we can say without doing any calculations that the domain of their composition will also be the set of all real numbers.

But if values are excluded from the domain of one function or another, things can get tricky.

Example 1

For example if f(x) = x^2 + 1 and g(x) = √(x-3) the domain of f(g(x)) will be obtained as follows. First we have to find all x in the domain of g. The domain of the square root function is the set of non-negative numbers. So x-3 needs to be greater than or equal to 0. And x is greater than or equal to 3. f(x) =x^2 + 1 is a polynomial. with a domain of all real numbers. So nothing prevents g(x) from being in the domain of f.

So putting that all together, the domain of f(g(x)) will be the set of x≥3

Example 2

To take a different example:

Suppose f(x) = 1/(x+2) and g(x) = 1/x

Again, the domain of y = f (g(x)) consists of all x which are in the domain of g such g(x) is also in the domain of f.

The domain of g is the set of x not equal to zero. The domain of f is the set of x not equal to -2. So the domain of f(g(x)) will consist of all x other than zero such that g(x) is not equal to -2. So we will solve g(x) = -2 to determine that excluded value...

1/x= -2 we can take the reciprocal of both sides

x= -1/2

Note that when x= -1/2, g(x) = -2 which is not in the domain of f.

Putting all this together, the domain of f(g(x)) will be the set x except for 0 and -1/2