Question

do f\left(x\right)=\sqrt{4x} and g\left(x\right)=\sqrt[3]{4x} have the same range

Answer 1

[AI-generated answer — Bot GPT 3.5]

To determine if the functions have the same range, we need to find the possible output values for each function.

For f(x) = √(4x), we note that the domain is restricted to x ≥ 0 because we cannot take the square root of a negative number in the real numbers. The range of f(x) will consist of all non-negative real numbers since the square root of a non-negative number is always non-negative. Therefore, the range of f(x) is [0, ∞).

For g(x) = ∛(4x), the domain is also restricted to x ≥ 0 for the same reason. The range of g(x) will consist of all real numbers since taking the cube root of any real number will always result in a real number. Therefore, the range of g(x) is (-∞, ∞).

Since the two functions have different ranges, f(x) = √(4x) and g(x) = ∛(4x) do not have the same range.

Answer 2

do they have the same domain?

Answer 3

[AI-generated answer — Bot GPT 3.5]

Yes, both functions f(x) = √(4x) and g(x) = ∛(4x) have the same domain of x ≥ 0 because the square root and cube root functions are defined for non-negative real numbers only.

Answer 4

are they increasing or decreasing?

Answer 5

[AI-generated answer — Bot GPT 3.5]

To determine if the functions f(x) = √(4x) and g(x) = ∛(4x) are increasing or decreasing, we need to analyze the slopes of the functions.

For f(x) = √(4x), we differentiate with respect to x. Using the power rule, we have:

f'(x) = (1/2) * (4x)^(-1/2) * 4
= 2 / √(4x)

Since the derivative f'(x) = 2 / √(4x) is always positive for x > 0, the function f(x) = √(4x) is increasing for x > 0.

For g(x) = ∛(4x), we differentiate with respect to x. Using the power rule, we have:

g'(x) = (1/3) * (4x)^(-2/3) * 4
= 4 / (3 * ∛(4x^2))

Again, the derivative g'(x) = 4 / (3 * ∛(4x^2)) is always positive for x > 0, so the function g(x) = ∛(4x) is also increasing for x > 0.

Therefore, both f(x) = √(4x) and g(x) = ∛(4x) are increasing functions for x > 0.