Question

Using the properties of combinations of continuous functions, determine the interval(s) over which the function f(x)=5x⁴4√x−18 continuous?

A) (0,[infinity])
B) (0,1)
C) [0,[infinity])
D) (−[infinity],[infinity])

Answer 1

The function f(x) = 5x⁴ √ x - 18 is continuous on the interval [0, ∞) because the fourth root is defined for x >= 0, and a polynomial is continuous everywhere.

Step-by-step explanation:

The function in question, f(x) = 5x⁴ √ x - 18, is a polynomial function with a radical component. To determine the intervals of continuity, we must first ensure the function is defined for all x in the interval. Since we have a fourth root (4√ x), the function is defined for x >= 0 because the fourth root of a negative number is not a real number. The polynomial part of the function (5x⁴) is continuous for all real numbers. Therefore, the function f(x) is continuous for every x where it is defined. The critical point to consider is x = 0 due to the fourth root; however, since the fourth root of 0 is 0, this causes no issues for continuity. Hence, the function is continuous from 0 to infinity, including 0. Hence the correct answer is option C: [0, ∞).