Functions (notation and composition)

Overview

Welcome to the study of Functions, a cornerstone of advanced mathematics. In OCR GCSE Further Mathematics, this topic (specification reference 2.5) moves beyond simple substitution and into the powerful concepts of composite and inverse functions. Think of a function as a precise recipe: it takes an ingredient (an input, x), follows a set of instructions, and produces a result (an output, f(x)). This guide will show you how to combine these recipes (composite functions) and how to work backwards from the result to the original ingredient (inverse functions). Mastery of this topic is not just about algebraic skill; it is about logical problem-solving. Examiners frequently use function questions to test a candidate's ability to handle abstract concepts and apply multi-step processes. Expect to see questions asking you to find fg(x), solve equations like gf(x) = 12, or determine the inverse function f⁻¹(x). This topic has strong synoptic links to solving equations, algebraic manipulation, and graph transformations.

Key Concepts

Concept 1: Function Notation

Function notation is the language we use to describe these mathematical machines. When you see f(x) = 2x + 5, it means the function f takes an input x, multiplies it by 2, and then adds 5. The letter f is just a name; you could have g(x), h(t), or any other letter. The variable in the bracket is the input.

Example: If f(x) = x² - 4, to find f(3), we substitute 3 for every x in the expression. So, f(3) = (3)² - 4 = 9 - 4 = 5. Credit is given for the correct substitution and evaluation.

Concept 2: Composite Functions

A composite function is a 'function of a function'. It involves applying one function to the output of another. The notation fg(x) means you apply the function g first, and then apply the function f to the result of g(x). A common mistake is to multiply f(x) by g(x), which is incorrect and will score no marks.

Example: Let f(x) = 2x and g(x) = x + 3. To find fg(x), we substitute the entire expression for g(x) into f(x).
fg(x) = f(g(x)) = f(x + 3) = 2(x + 3) = 2x + 6.
To find gf(x), we do the reverse, substituting f(x) into g(x):
gf(x) = g(f(x)) = g(2x) = 2x + 3.
Notice that fg(x) is not the same as gf(x). The order is critical.

Concept 3: Inverse Functions

An inverse function, denoted f⁻¹(x), reverses the action of the original function. If f turns a into b, then f⁻¹ will turn b back into a. A crucial point is that f⁻¹(x) is not the same as 1/f(x) (the reciprocal). This is a common misconception that examiners see frequently.

To find the inverse of a function, we use a reliable three-step method:

1. Let y = f(x): Rewrite the function with y as the subject.
2. Rearrange for x: Make x the subject of the new equation.
3. Swap x and y: Replace every y with an x and the x with f⁻¹(x) to state the final inverse function.

Example: Find the inverse of f(x) = 3x - 2.
Step 1: Let y = 3x - 2.
Step 2: Rearrange for x. y + 2 = 3x, so x = (y + 2) / 3.
Step 3: Swap x and y. y = (x + 2) / 3. So, f⁻¹(x) = (x + 2) / 3.

Mathematical/Scientific Relationships

• Function Definition: A rule that assigns to each input exactly one output. y = f(x)
• Composite Function: fg(x) = f(g(x)) (Apply g, then f). Must memorise.
• Inverse Function: f⁻¹(f(x)) = x and f(f⁻¹(x)) = x. The inverse function undoes the original function. Must memorise.
• Finding an Inverse: The three-step process (Let y = ..., Rearrange for x, Swap x and y). This is a method, not a formula, and must be memorised.

Practical Applications

While functions in GCSE Further Maths are largely abstract and algebraic, the underlying concepts are fundamental to many real-world systems. For example, in computer programming, functions are blocks of code that take an input and return an output, and chaining them together is a form of composition. In economics, functions can model cost, revenue, and profit based on production levels. In physics, equations of motion are functions of time. Understanding how to manipulate and combine these functions is a foundational skill for STEM fields.