Mathematics - Additional (0606) | Functions

📚 Additional Mathematics (0606) Study Notes: Functions

👋 Hello and Welcome to the World of Functions!

Functions are one of the most foundational and important topics in Additional Mathematics. Don't worry if the notation looks scary at first; a function is just a special type of machine that takes an input, follows a rule, and spits out exactly one output.

Mastering this chapter is vital because functions appear everywhere—from sketching complex graphs to understanding calculus. Let's break down the essential concepts step-by-step!

1. Defining a Function: The Machine Analogy (Syllabus 1.1)

1.1 What Exactly is a Function?

A function, \(f\), is a rule that assigns every element in a starting set (the input) to exactly one element in an ending set (the output).

The standard notation used is:

• Function Notation: \(f(x) = 3x + 2\). This means "the function of x is 3x plus 2."
• Mapping Notation: \(f: x \to 3x + 2\). This reads "the function f maps x onto 3x plus 2."

1.2 Key Terminology: Domain and Range (Syllabus 1.1, 1.2)

When dealing with functions, we need to define which numbers are allowed to go in and which numbers come out.

1. Domain (The Input Set)

• The Domain is the complete set of all possible input values (\(x\)) for which the function is defined.
• Analogy: These are the ingredients you are allowed to put into the machine.

2. Range (The Output Set or Image Set)

• The Range is the complete set of all possible output values (\(f(x)\) or \(y\)) that the function produces when using the defined domain.
• Analogy: These are the products that come out of the machine.

How to Find the Domain and Range (Identifying Restrictions)

Unless specified otherwise, the domain of most functions is all real numbers, \(x \in \mathbb{R}\). However, you must look out for two main restrictions:

Restriction A: Division by Zero
If the function is a fraction, the denominator cannot be zero.
Example: For \(f(x) = \frac{1}{x-4}\), the domain is \(x \in \mathbb{R}, x \ne 4\).

Restriction B: Square Roots of Negative Numbers
If the function involves a square root, the expression under the root sign must be greater than or equal to zero.
Example: For \(g(x) = \sqrt{x+5}\), we need \(x+5 \ge 0\), so the domain is \(x \ge -5\).

Finding the Range:
Often, the easiest way to determine the range is by sketching the graph or by considering the maximum/minimum points (especially for quadratic functions, Topic 2).
Example: For \(h(x) = x^2 + 1\), since \(x^2 \ge 0\) for all real \(x\), the smallest output is \(0+1=1\). The range is \(h(x) \ge 1\).

2. The Modulus Function (Absolute Value) (Syllabus 1.4)

The modulus function, written as \(y = |f(x)|\), gives the absolute value of the output. This means the result is never negative.

2.1 Graphing \(y = |f(x)|\)

When you sketch the graph of a modulus function, the process is simple:

1. First, sketch the graph of the original function, \(y = f(x)\).
2. Any part of the graph that is above or on the x-axis remains exactly the same.
3. Any part of the graph that is below the x-axis (where \(y\) is negative) must be reflected upwards across the x-axis.

The range of \(y = |f(x)|\) will always be \(y \ge 0\), or greater than or equal to the minimum point if the original graph had a minimum above zero.

Did you know? The sharp points created when you reflect the graph up are called cusps. These are important features to label when sketching.

3. Function Types and Inverses (Syllabus 1.1, 1.5, 1.6, 1.8)

Not all functions are created equal! To find an inverse, a function must be a special type called a one-one function.

3.1 One-one vs. Many-one Functions

We use the Horizontal Line Test (HLT) to distinguish between types of functions:

1. One-one Function

• Each output corresponds to only one input.
• Test: A horizontal line crosses the graph at most once.
• Inverse: An inverse function exists.

2. Many-one Function (Syllabus 1.5)

• At least one output corresponds to two or more inputs.
• Test: A horizontal line can cross the graph more than once (e.g., a standard parabola \(y=x^2\)).
• Inverse: An inverse function does NOT exist unless the domain is restricted to make it one-one.

3.2 Finding the Inverse Function, \(f^{-1}(x)\) (Syllabus 1.6)

The inverse function, \(f^{-1}(x)\), reverses the mapping. The domain of \(f\) becomes the range of \(f^{-1}\), and vice-versa.

Step-by-Step Method for Finding \(f^{-1}(x)\):

1. Step 1: Write \(y = f(x)\).
   Example: If \(f(x) = 2x - 3\), write \(y = 2x - 3\).
2. Step 2: Swap \(x\) and \(y\). (This is the crucial inversion step!)
   Example: \(x = 2y - 3\).
3. Step 3: Rearrange to make \(y\) the subject.
   Example: \(x + 3 = 2y \implies y = \frac{x+3}{2}\).
4. Step 4: Rewrite using inverse notation.
   Example: \(f^{-1}(x) = \frac{x+3}{2}\).

3.3 The Graphical Relationship (Syllabus 1.8)

The graph of a function \(y = f(x)\) and its inverse \(y = f^{-1}(x)\) have a simple and beautiful relationship:

They are reflections of each other in the line \(y = x\).

If a point \((a, b)\) lies on \(f(x)\), then the point \((b, a)\) lies on \(f^{-1}(x)\). This is why we swap \(x\) and \(y\) algebraically!

4. Composite Functions (Syllabus 1.1, 1.7)

A composite function is formed when one function is applied to the result of another function. Think of it as linking two function machines together.

4.1 Notation and Order

If we have two functions \(f(x)\) and \(g(x)\):

• The composite function \(fg(x)\) means apply \(g\) first, then apply \(f\) to the result.
  We write this as \(f(g(x))\).
• The composite function \(gf(x)\) means apply \(f\) first, then apply \(g\) to the result.
  We write this as \(g(f(x))\).

IMPORTANT: The order matters! In general, \(fg(x)\) is NOT the same as \(gf(x)\) (Syllabus 1.7).

The "Inside-Out" Rule: Always substitute the 'inner' function into the 'outer' function.

Example: Let \(f(x) = x + 1\) and \(g(x) = x^2\).
To find \(fg(x)\): Substitute \(g(x)\) into \(f(x)\).
\(fg(x) = f(x^2) = (x^2) + 1\).
To find \(gf(x)\): Substitute \(f(x)\) into \(g(x)\).
\(gf(x) = g(x + 1) = (x + 1)^2\).

4.2 Composition of a Function with Itself: \(f^2(x)\) (Syllabus 1.3)

The notation \(f^2(x)\) means the function \(f\) composed with itself: \(f(f(x))\).

Example: Let \(f(x) = 2x + 5\).
\(f^2(x) = f(f(x)) = f(2x + 5) = 2(2x + 5) + 5\)
\(f^2(x) = 4x + 10 + 5 = 4x + 15\).

4.3 Domain and Range of Composite Functions (Syllabus 1.2)

Finding the domain and range of composite functions can be tricky, especially if the original functions have restricted domains.

For \(fg(x)\):
1. The Domain of \(fg\) is the set of inputs (\(x\)) allowed by the inner function (\(g\)). (Domain \(fg \subseteq\) Domain \(g\)).
2. The Range of \(fg\) is the set of outputs (\(y\)) produced by the outer function (\(f\)) when fed the outputs of \(g\). (Range \(fg \subseteq\) Range \(f\)).

Encouragement: The key is to check that the outputs of the first function are valid inputs for the second function. If the range of \(g\) is not suitable for the domain of \(f\), then the domain of \(g\) must be restricted for \(fg\) to exist.

Summary Checklist: Essential Skills

You are ready for the exam on this topic if you can confidently:

• Define function, domain, range, one-one, and inverse function.
• Identify restrictions on the domain (division by zero, square roots).
• Use function notation: \(f(x)\), \(f: x \to \dots\), \(f^{-1}(x)\), \(fg(x)\), \(f^2(x)\).
• Form composite functions \(fg\) and \(gf\), remembering the order is critical.
• Explain why a many-one function (like a parabola) does not have an inverse.
• Find the inverse function \(f^{-1}(x)\) using the swap and rearrange method.
• Sketch the graph of \(y = |f(x)|\) by reflecting the negative parts upward.
• Show graphically that \(f\) and \(f^{-1}\) are reflections in the line \(y = x\).