Final answer:
Using the horizontal line test, the function f(x)=2x³-4x is not one-to-one since horizontal lines intersect the graph at multiple points over its domain of all real numbers.
Step-by-step explanation:
To determine whether the function f(x)=2x³-4x is one-to-one, we should use the horizontal line test. This test involves drawing horizontal lines across the graph of the function and checking if any horizontal line intersects the graph more than once. If any horizontal line does cross the graph more than once, the function is not one-to-one because it would mean that a single y-value is paired with multiple x-values, violating the definition of a one-to-one function.
In the case of the cubic function f(x)=2x³-4x, if we graph it, we will notice that it's a smoothly curving line with no abrupt changes in direction. Performing the horizontal line test, you will find that for certain ranges of y-values, such lines will intersect the graph at three points, indicating that f(x) is not one-to-one. This is because the graph of f(x) is not strictly increasing or decreasing; it has both a local maximum and minimum within the domain of all real numbers.
However, if we were to restrict the domain of f(x), we might be able to find an interval where the function becomes one-to-one. But for all real numbers, as stated in the function without restrictions, the horizontal line test confirms that the function is not one-to-one.