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.

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.