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CIE A-Level Maths Study Notes

1.2.2 Types of Functions

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

Example:

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

  • Let f(a)=f(b)f(a) = f(b).
  • Then, 2a+3=2b+32a + 3 = 2b + 3.
  • Subtracting 3 from both sides gives 2a=2b2a = 2b.
  • Dividing by 2 gives a=ba = b, proving that f(x)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)=x2f(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)=x2f(x) = x^2 is not one-one over the set of all real numbers.
  • However, by restricting the domain to x0x ≥ 0, the horizontal line will intersect the graph at most once, making the restricted function one-one.
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Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
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Oxford University - PhD Mathematics

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.

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