In mathematics, a function is one-to-one (or injective) if it maps distinct elements in its domain to distinct elements in its range. That is, for any two different elements a and b in the domain, the corresponding function values f(a) and f(b) must also be different. A function can be shown to be one-to-one using algebraic methods, which involve manipulating equations and inequalities to prove that the function satisfies the one-to-one property.

How does one show that a function is one-to-one using algebra?

Using the Definition of a One-to-One Function

The simplest way to show that a function f is one-to-one is to use the definition directly. That is, we assume that f(a) = f(b) for some elements a and b in the domain, and then show that a = b must be true. This can be done by applying algebraic techniques to manipulate the equation f(a) = f(b) and simplify it until we obtain the desired result.

For example, consider the function f(x) = x^3 defined on the real numbers. To show that f is one-to-one, suppose that f(a) = f(b) for some a and b in the domain. Then we have:

a^3 = b^3
Taking the cube root of both sides gives:

a = b
Therefore, f is one-to-one, since the only way for f(a) = f(b) to be true is if a = b.

Using Algebraic Manipulations

Another way to show that a function is one-to-one is to use algebraic manipulations to simplify the function and then apply inequalities to prove that the function is increasing or decreasing on its entire domain. If the function is increasing or decreasing, then it must be one-to-one.

For example, consider the function f(x) = 2x – 3 defined on the real numbers. To show that f is one-to-one, we can first simplify the function by factoring out the 2:

f(x) = 2(x – 3/2)
Now, notice that f(x) is increasing on its entire domain, since adding a positive value to x increases the value of f(x) by 2. To prove this formally, we can use the following inequality:

f(x + h) – f(x) = 2[(x + h) – 3/2] – 2(x – 3/2) = 2h > 0
Since h is positive, it follows that f(x + h) > f(x) for all x, which implies that f is one-to-one.

Using the Derivative Test

A third method for showing that a function is one-to-one is to use the derivative test, which involves finding the derivative of the function and then analyzing its sign on the entire domain. If the derivative is positive or negative on the entire domain, then the function is increasing or decreasing, respectively, and hence one-to-one.

For example, consider the function f(x) = x^2 defined on the real numbers. To show that f is one-to-one, we can take its derivative and analyze its sign:

f'(x) = 2x
The derivative is positive for all x > 0, negative for all x < 0, and zero only at x = 0. Therefore, f is increasing on (0, ∞) and decreasing on (-∞, 0). Since f is continuous, it follows that f is one-to-one on its entire domain.

Conclusion

In conclusion, there are several algebraic methods for showing that a function is one-to-one. The simplest approach is to use the definition directly and show that the function maps distinct elements to distinct elements. Alternatively, one can use algebraic manipulations to simplify the function and then apply inequalities to prove that the function is increasing or decreasing on its entire domain. Finally, the derivative test can be used to analyze the sign of the derivative and show that the function is one-to-one. Regardless of the method used, algebraic techniques provide a powerful tool for analyzing the properties of functions and establishing their one-to-one property.