Find the domain of the composite function
fog.
f(x)=x^2+3;g(x)=sqrtx-4
Consider the function F(X) = X2+5 and G(X) = √x . FIND THE COMPOSITE FUNTION (f o g)(x) and also find the domain of this composite function.
To find the domain of the composite function f o g, we need to determine the set of all possible values of x that can be input into the composite function and give a valid output.
According to the given information,
f(x) = x^2 + 3, and
g(x) = √x - 4.
To find the composite function f o g, we need to substitute g(x) into f(x):
f o g(x) = f(g(x)) = f(√x - 4).
To find the domain of f o g, we need to consider two things:
1. The domain of g(x), which represents the values of x that can be input into g(x) without resulting in an undefined or complex output.
2. The domain of f, which represents the values of g(x) that can be input into f(x) without resulting in an undefined or complex output.
Let's start with finding the domain of g(x):
g(x) = √x - 4.
For the square root function (√x) to be defined, the radicand (x) must be greater than or equal to zero. So, we have:
x ≥ 0.
Next, we need to find the domain of f(g(x)). To determine this, we need to consider the domain of f(x):
f(x) = x^2 + 3.
The function f(x) is defined for all real numbers, so there are no restrictions on the inputs.
Now, we substitute g(x) into f(x):
f(g(x)) = f(√x - 4) = (√x - 4)^2 + 3.
The square of any real number is always non-negative, so (√x - 4)^2 will always be greater than or equal to zero.
Considering the fact that the square of any real number is always non-negative and adding the fact that there are no restrictions on the domain of f(x), the domain of f o g is:
x ≥ 0.
Therefore, the domain of the composite function f o g is all x-values greater than or equal to zero.