Question

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

Answer 1

[AI-generated answer — Bot GPT 3.5]

Yes, the functions f(x) = -√x and g(x) = -∛x have the same domain. The domain of both functions is all real numbers greater than or equal to 0.

Answer 2

are they both decreasing?

Answer 3

[AI-generated answer — Bot GPT 3.5]

Yes, both functions are decreasing. As x increases, the values of -√x and -∛x also decrease.

Answer 4

are they both increasing?

Answer 5

[AI-generated answer — Bot GPT 3.5]

Apologies for the confusion. Neither of the functions f(x) = -√x or g(x) = -∛x are increasing.

The function f(x) = -√x is a decreasing function because as x increases, the value of -√x decreases.

The function g(x) = -∛x is also a decreasing function because as x increases, the value of -∛x decreases.

In summary, neither function is increasing; both are decreasing.

Answer 6

do they have the same range?

Answer 7

[AI-generated answer — Bot GPT 3.5]

No, the functions f(x) = -√x and g(x) = -∛x do not have the same range.

The range of f(x) = -√x is all real numbers less than or equal to 0, since the square root of a non-negative number is always non-negative and we take the negative of that value. So, the range of f(x) is (-∞, 0].

On the other hand, the range of g(x) = -∛x is all real numbers, since the cube root function can take both positive and negative values. So, the range of g(x) is (-∞, +∞).

Therefore, the ranges of the two functions are different.

Answer 8

are they both decreasing, or do they both have the same domain?

Answer 9

[AI-generated answer — Bot GPT 3.5]

Apologies for the confusion in my previous response. Let's clarify:

Both functions f(x) = -√x and g(x) = -∛x are decreasing functions, as the values of f(x) and g(x) decrease as x increases.

They also have the same domain, which is all real numbers greater than or equal to 0. Both functions are defined for non-negative real numbers, as the square root (√) and cube root (∛) functions are defined for non-negative real numbers.