How do you find the domain and range of #f(x) = x^3 - 3x + 2#?

Question

Kennedy Adams · Accepted Answer

#D: " "(-oo, +oo)#

#R: " "(-oo, + oo)#

Domain simply asks, "Where does the function exist on the x-axis?" So, in this function, because #x# is cubed, it can stretch from negative infinity to positive infinity.

Similarly, range asks, "Where does the function exist on the y-axis?" When you plug the function into a graph, it becomes evident that it will forever go upward toward infinity and forever downwards toward negative infinity on both axes.

This image shows the basic graph of #f(x)=x^3#.

Savannah Anderson · Answer

undefined To find the domain of $ f(x) = x^3 - 3x + 2 $, we need to determine all possible values of $ x $ for which the function is defined. Since $ f(x) $ is a polynomial function, it is defined for all real numbers. Therefore, the domain of $ f(x) $ is $ \mathbb{R} $, the set of all real numbers.

To find the range of $ f(x) = x^3 - 3x + 2 $, we can analyze the behavior of the function as $ x $ approaches positive and negative infinity. As $ x $ approaches positive infinity, $ x^3 $ dominates the expression, leading to a positive value. Similarly, as $ x $ approaches negative infinity, $ x^3 $ dominates the expression, leading to a negative value. Therefore, the range of $ f(x) $ is $ \mathbb{R} $, the set of all real numbers.