Answer:
the correct answers are
f(x) = x^3 + 2,
f(x) = x^2 + 2,
and f(x) = 3x + 4.
Explanation:
The following functions are increasing throughout their domains:
f(x) = x3 + 2
f(x) = x2 + 2
f(x) = 3x + 4
We can use the derivative to determine whether a function is increasing or decreasing. The derivative of a function at a point gives the slope of the function's graph at that point. If the derivative is positive, the slope is positive and the function is increasing. If the derivative is negative, the slope is negative and the function is decreasing.
The derivatives of the given functions are:
f'(x) = 3x2
f'(x) = 2x
f'(x) = 3
All of these derivatives are always positive for any real number x. Therefore, the corresponding functions are increasing throughout their domains.
The following functions are not increasing throughout their domains:
f(x) = -x3
f(x) = -4x
The derivative of f(x) = -x3 is f'(x) = -3x2, which is negative for x > 0 and positive for x < 0. Therefore, f(x) is decreasing for x > 0 and increasing for x < 0.
The derivative of f(x) = -4x is f'(x) = -4, which is always negative. Therefore, f(x) is always decreasing.
Therefore, the correct answer is f(x) = x3 + 2, f(x) = x2 + 2, and f(x) = 3x + 4.