Question

For the real-valued functions f(x)=x²-3 and g(x)=√x 2, find the composition f∘g and specify its domain using interval notation.

(f∘g)(x)=___________

Domain of f∘g=_______

Answer 1

The composition (f∘g)(x) is x - 3 and its domain is [0, ∞). We obtain this by substituting √x for x in f(x) and taking into account the domain of g(x) which only allows non-negative x values.

Step-by-step explanation:

To find the composition f∘g and its domain, we need to substitute g(x) into f(x). The function g(x) = √x can only be evaluated for x≥0 since we can't take the square root of a negative number in the real number system. So, g(x) will provide non-negative outputs.

Now let's substitute g(x) into f(x) to get (f∘g)(x):
f(g(x)) = f(√x) = (√x)^2 - 3 = x - 3.

The domain of f(x) = x^2 - 3 is all real numbers because any real number squared is a real number. However, because we started with g(x) = √x, we need to consider the domain of g(x) which is [0, ∞). Therefore, the domain of the composition f∘g is also [0, ∞) since that's the range of inputs for which g(x) = √x is defined, and then can subsequently be plugged into f(x) = x^2 - 3.