Functions are at the heart of A-Level Maths, serving as a bridge between sets of numbers or objects. They are defined by the relationship that assigns to each element of a set, called the domain, exactly one element of another set, known as the codomain. A deeper understanding of one-one functions is essential as it sets the stage for exploring more complex mathematical concepts.

One-One Functions (Injective Functions)

Definition

A one-one function, or an injective function, ensures a unique mapping: for every element in the domain, there is a unique element in the codomain. This means no two distinct elements of the domain map to the same element of the codomain.

Properties

• Distinct Output: Every x-value in the domain is paired with a distinct y-value in the codomain.
• No Duplicates: No y-value in the codomain is the image of more than one x-value from the domain.

Testing for One-One Functions

There are two primary methods to test for one-oneness:

1. Algebraic Test:

• Consider a function $f(x)$.
• Assume $f(a) = f(b)$ for some a and b in the domain.
• Show that $a = b$. If you can, then the function is one-one.

Example:

Prove that $f(x) = 2x + 3$ is one-one.

• Let $f(a) = f(b)$.
• Then, $2a + 3 = 2b + 3$.
• Subtracting 3 from both sides gives $2a = 2b$.
• Dividing by 2 gives $a = b$, proving that $f(x)$ is one-one.

2. Graphical Test (Horizontal Line Test):

• Draw the graph of the function.
• If every horizontal line y = k (for any constant k) intersects the graph at no more than one point, the function is one-one.

Example:

Consider the function $f(x) = x^2$.

• Drawing the graph, we see that horizontal lines intersect the graph in two points for y-values greater than zero.
• Therefore, $f(x) = x^2$ is not one-one over the set of all real numbers.
• However, by restricting the domain to $x ≥ 0$, the horizontal line will intersect the graph at most once, making the restricted function one-one.