Understanding One-to-One Functions: The Case of (f(x) x^6)

In the context of mathematical functions, particularly in the realm of calculus and advanced algebra, one-to-one functions, also known as injective functions, play a crucial role. A function (f(x)) is considered one-to-one if each input (x) is paired with exactly one output (f(x)), and each output (f(x)) is paired with exactly one input (x). This article will delve into the specific case of (f(x) x^6) to illustrate the concept.

Definition of One-to-One Functions

Formally, a function (f(x)) is one-to-one if whenever (f(a) f(b)), it must follow that (a b).

Verification Using the Definition

To show that (f(x) x^6) is a one-to-one function, we start by assuming (f(a) f(b)), where (a) and (b) are distinct inputs in the domain of the function.

Assume (f(a) f(b)). This implies (a^6 b^6). Subtract (b^6) from both sides to get (a^6 - b^6 0). Factor the difference of cubes: ((a^3 - b^3)(a^3 b^3) 0). Since (a^3 - b^3 0) and (a^3 b^3 eq 0) for real numbers (a) and (b), we must have (a b). Thus, whenever (f(a) f(b)), it follows that (a b).

This confirms that (f(x) x^6) is indeed a one-to-one function.

Horizontal Line Test

An alternative method to verify that a function is one-to-one is to use the horizontal line test. The horizontal line test states that if any horizontal line intersects the graph of the function at most once, then the function is one-to-one.

For the function (f(x) x^6):

This is a polynomial function with an even degree (6), specifically a monomial function. The coefficient of (x^6) is positive, making the graph rise continuously as (x) moves away from zero in either direction. A horizontal line will intersect this parabolic curve at most once for any value of (y).

Hence, (f(x) x^6) passes the horizontal line test, confirming its one-to-one nature.

One-to-One by Definition

By definition, a function (f(x)) is one-to-one if for each (y), there is at most one (x) such that (f(x) y).

Given (y x^6), solving for (x) gives (x y^{1/6}) or (x -y^{1/6}), but since (x^6) is always non-negative, (x y^{1/6}) is the only solution for each (y). Thus, for each (y), there is exactly one (x) such that (f(x) y) and hence (f) is one-to-one.

Further Clarification

The function (f(x) x^6) is a one-to-one function because for every value of (x), there is exactly and only one value of (f(x)) and for every value of (f(x)), there is exactly and only one value of (x).

This property holds for any straight line that is neither vertical nor horizontal. The graph of (f(x) x^6) is a straight line (specifically a power function) that is neither vertical nor horizontal, confirming its one-to-one nature.

It is important to note that when the exponent is 2 (or any even number), as in (f(x) x^2 - 6), the function is not one-to-one because more than one (x) value can correspond to a single (y) value, violating the one-to-one condition.

In Conclusion

To summarize, (f(x) x^6) is a one-to-one function because it satisfies the condition that each input maps to exactly one output and vice versa. This characteristic is confirmed both algebraically and graphically through the horizontal line test. Understanding one-to-one functions is essential in many areas of mathematics, including calculus and algebra, and helps establish the uniqueness of solutions in mathematical models.