Functions Quizzes

Introduction

Functions are a fundamental concept in GCSE Maths, forming the basis for understanding relationships between variables. Mastery of functions allows students to interpret and manipulate mathematical relationships, prepare for graphing, algebra, sequences, and higher-level topics. Functions appear frequently in both foundation and higher-tier exams, and understanding them is crucial for problem-solving and real-life applications such as physics, finance, and computer science.

Core Concepts

What is a Function?

A function is a rule that assigns exactly one output value for each input value. It can be thought of as a machine: for every input \(x\), there is a single output \(f(x)\).

Example:

$$ f(x) = 2x + 3 $$>

For each input \(x\), we multiply by 2 and then add 3 to get the output.

Function Notation

Functions are often written in the form \(f(x)\), \(g(x)\), or \(h(x)\). Examples:

• \(f(x) = x^2 - 4x + 5\)
• \(g(x) = 3x - 7\)
• \(h(x) = \frac{1}{x}\)

Here, \(x\) is the input variable, and \(f(x)\) is the output.

Domain and Range

• Domain: The set of all possible input values \(x\).
• Range: The set of all possible output values \(f(x)\).

Example: \(f(x) = \sqrt{x}\)

• Domain: \(x \geq 0\) (cannot take the square root of a negative number)
• Range: \(f(x) \geq 0\)

Rules & Steps

1. Evaluating Functions

To find \(f(a)\):

1. Replace every \(x\) in the function with \(a\).
2. Simplify to find the output.

Example:

$$ f(x) = 2x + 5, \quad f(3) = 2(3) + 5 = 11 $$

2. Function Composition

Function composition involves applying one function to the result of another: \((f \circ g)(x) = f(g(x))\).

Example:

$$ f(x) = x + 2, \quad g(x) = 3x $$ $$ (f \circ g)(x) = f(g(x)) = f(3x) = 3x + 2 $$

3. Inverse Functions

The inverse function reverses the effect of the original function. Denoted \(f^{-1}(x)\).

1. Replace \(f(x)\) with \(y\).
2. Swap \(x\) and \(y\).
3. Solve for \(y\).

Example:

$$ f(x) = 2x + 3 $$ $$ y = 2x + 3 \Rightarrow x = 2y + 3 \quad (\text{swap } x \text{ and } y) $$ $$ x = 2y - 3 \Rightarrow y = \frac{x - 3}{2} $$ $$ f^{-1}(x) = \frac{x - 3}{2} $$

4. Graphing Functions

• Linear functions (\(f(x) = mx + c\)) → straight lines with slope \(m\) and y-intercept \(c\)
• Quadratic functions (\(f(x) = ax^2 + bx + c\)) → parabolas
• Other functions, e.g., \(f(x) = \frac{1}{x}\), \(f(x) = \sqrt{x}\), → curves with specific properties

Worked Examples

1. Evaluate \(f(x) = x^2 - 4x + 6\) at \(x = 3\): $$ f(3) = 3^2 - 4(3) + 6 = 9 - 12 + 6 = 3 $$
2. Function composition: \(f(x) = x + 1\), \(g(x) = 2x\), find \((g \circ f)(x)\): $$ (g \circ f)(x) = g(f(x)) = g(x + 1) = 2(x + 1) = 2x + 2 $$
3. Inverse function: \(f(x) = 3x - 7\), find \(f^{-1}(x)\): $$ y = 3x - 7 \Rightarrow x = 3y - 7 \Rightarrow 3y = x + 7 \Rightarrow y = \frac{x + 7}{3} $$
4. Graph linear function \(f(x) = -2x + 4\):
   • y-intercept = 4, slope = -2
   • Plot points: (0,4), (1,2), (2,0)
   • Draw straight line through points
5. Quadratic function: \(f(x) = x^2 - 4x + 3\)
   • Factorise: \((x-1)(x-3)\)
   • Roots: \(x = 1, 3\)
   • Vertex at \(x = 2\), \(y = -1\), plot points and draw parabola opening upwards

Common Mistakes

• Confusing input and output; remember \(f(x)\) is the output for input \(x\).
• Errors in function composition; order matters (\(f \circ g \neq g \circ f\) in general).
• Incorrectly finding inverse functions; always swap x and y first.
• Graphing mistakes: forgetting slope, intercepts, or vertex positions.
• Assuming all functions are linear; check the type of function before graphing.

Applications

• Physics: velocity-time relationships
• Economics: cost, revenue, and profit functions
• Biology: population growth models
• Engineering: input-output systems
• Problem-solving: modeling real-world relationships and predictions

Strategies & Tips

• Always identify the type of function before evaluating or graphing.
• Check domains and ranges to ensure meaningful outputs.
• Use function composition to simplify complex problems step by step.
• When finding inverses, follow the step-by-step algebra carefully.
• Practice graphing functions by plotting multiple points to ensure accuracy.

Summary

Functions are a central concept in GCSE Maths, enabling students to interpret, evaluate, and manipulate relationships between variables. Understanding notation, composition, inverses, and graphing equips students for problem-solving and higher-level mathematics. Regular practice with evaluation, composition, and graphing exercises strengthens conceptual understanding and prepares students for exams. Attempt the quizzes and exercises to consolidate your knowledge of functions and enhance mathematical confidence.