Determining the Domain of f(g(x)) Where f(x) 3x^2 - 4x^2 and g(x) √(x^5)

The domain of a functionrsquo;s composition, f(g(x)), is defined as the set of x-values for which the function is well-defined. This involves identifying the individual domains of the component functions and then finding their intersection. Letrsquo;s explore how to determine the domain of f(g(x)) where f(x) 3x^2 - 4x^2 and g(x) √(x^5).

Understanding the Functions

First, we need to understand the individual functions:

Function f(x) 3x^2 - 4x^2:

This function can be simplified as f(x) -x^2, which is defined for all real numbers. Therefore, the domain of f(x) is all real numbers, denoted by {all real numbers}.

Function g(x) √(x^5):

The function g(x) involves the square root of x^5. For the square root to be defined, the radicand (the expression inside the square root) must be non-negative. Therefore, we need x^5 ≥ 0.

Since x^5 is non-negative if and only if x is non-negative (x ≥ 0) or x is a negative number to an odd power, we find that g(x) is defined for x ≥ 0.

Thus, the domain of g(x) is {x ≥ 0}.

Determining the Domain of f(g(x))

Now, we need to determine the domain of f(g(x)). To do this, we need to evaluate the composition f(g(x)). First, we substitute g(x) into f(x):

fg(x) f(g(x)) -[g(x)]^2 -[√(x^5)]^2 -x^5.

For this function to be defined, we need to ensure that the domain of g(x) is valid for f(g(x)).

Since the domain of g(x) is x ≥ 0, we need to check if this restriction is valid for f(g(x)). Since -x^5 is defined for all x ≥ 0, the domain of f(g(x)) is the intersection of the domains of f and g.

Therefore, the domain of f(g(x)) is {x ≥ 0}.

In conclusion, the problem of determining the domain of f(g(x)) involves understanding the domains of the individual functions and finding the intersection. For f(x) -x^2 (defined for all real numbers) and g(x) √(x^5) (defined for x ≥ 0), the domain of f(g(x)) is {x ≥ 0}.

Therefore, the domain of f(g(x)) is [-5, ∞).