How can you tell if a function is one-to-one?

To determine if a function is one-to-one (also known as injective), you need to check if each element of the function’s domain maps to a unique element in its codomain. In simpler terms, a function is one-to-one if no two different inputs produce the same output.

Here’s a step-by-step approach to check if a function is one-to-one:

1. Understand the Function: Begin by examining the function you want to analyze. It could be represented as a set of ordered pairs, a graph, or an algebraic expression.
2. Use the Horizontal Line Test: If you have a graph of your function, you can apply the horizontal line test. Draw horizontal lines across the graph; if any horizontal line intersects the graph more than once, the function is not one-to-one.
3. Analyze Algebraically: If you’re working with an algebraic function, solve the equation f(a) = f(b) where a and b are distinct elements in the domain. If you find that a = b is the only solution, then the function is one-to-one.
4. Consider the Inverse: Another way to think about one-to-one functions is by considering whether an inverse exists. If a function has an inverse that is also a function, then the original function is one-to-one.

For example, consider the function f(x) = 2x + 3. To see if it’s one-to-one, you can:

• Graph it to apply the horizontal line test (it passes since no horizontal line intersects it more than once).
• Set f(a) = f(b): 2a + 3 = 2b + 3. Simplifying gives 2a = 2b, and thus a = b.

Since both tests indicate that every input maps to a unique output, f(x) = 2x + 3 is indeed a one-to-one function.

In summary, you can determine if a function is one-to-one through graphical analysis, algebraic manipulation, and by considering if an inverse function exists. This understanding is crucial for various mathematical applications, including solving equations and understanding function behavior.