Mathematics - Analysis and Approaches | Functions

🌟 Welcome to Functions: The Foundation of Analysis! 🌟

Hello future IB mathematicians! This chapter, Functions, is arguably the most crucial unit in Mathematics Analysis and Approaches. Why? Because functions are the bedrock upon which Calculus, Modeling, and advanced algebra are built. Mastering this topic gives you the analytical framework necessary to truly understand how mathematical relationships work—the core focus of the AA course.

Don't worry if some concepts, like finding complex domains or inverse functions, seem challenging initially. We will break down every idea into clear, manageable steps. Let's get started on understanding how things relate!

1. Defining Functions: Notation, Domain, and Range

1.1 What Exactly is a Function?

In simple terms, a function is a rule that assigns exactly one output for every input. It’s like a vending machine: you press one specific button (input), and you get one specific snack (output). You never press 'A1' and sometimes get a chocolate bar and sometimes get chips.

• Input: The variable, usually \(x\), that you put into the function.
• Output: The resulting variable, usually \(y\) or \(f(x)\).

We use the notation \(f(x)\) which means "the function \(f\) applied to the input \(x\)".

Key Terminology: Domain and Range

The mathematical rules governing functions require us to be precise about what inputs are allowed and what outputs are possible.

1. The Domain (The Inputs):

• The Domain is the complete set of all possible input values (\(x\)-values) for which the function is defined.
• Analogy: If a function is a recipe, the domain is the list of ingredients you are allowed to use.

2. The Range (The Outputs):

• The Range is the complete set of all resulting output values (\(y\)-values or \(f(x)\) values) that the function can produce.
• Analogy: The range is the set of all dishes that can possibly be made from your recipe.

🚨 Common Mistake Alert: Identifying Restrictions (HL Focus on Rigour) 🚨

When asked to find the domain of a function, you are looking for inputs that would cause a mathematical error. There are two primary "no-go" zones:

1. Division by Zero: The denominator of any fraction cannot equal zero.

   Example: For \(f(x) = \frac{1}{x-3}\), we must have \(x-3 \neq 0\), so \(x \neq 3\). The domain is \(\{x \in \mathbb{R} \mid x \neq 3\}\).

2. Even Roots of Negative Numbers: The expression underneath a square root, fourth root, etc., must be greater than or equal to zero.

   Example: For \(g(x) = \sqrt{2x+8}\), we must have \(2x+8 \ge 0\), so \(x \ge -4\). The domain is \(\{x \in \mathbb{R} \mid x \ge -4\}\).

2. Visualizing Functions: Graphs and the Line Tests

2.1 The Vertical Line Test (VLT)

We use the graph of a relationship to determine visually if it is a function.

• If you can draw a vertical line anywhere on the graph that crosses the graph at more than one point, then that relationship is NOT a function. (This means one input \(x\) has multiple outputs \(y\).)

Example: A circle fails the VLT, so it is not the graph of a function. A parabola opening upwards passes the VLT.

2.2 Function Classification (One-to-One vs. Many-to-One)

Functions can be classified based on how inputs map to outputs.

1. Many-to-One:
Multiple different inputs can lead to the same output.
Example: \(f(x) = x^2\). \(f(2) = 4\) and \(f(-2) = 4\). This is acceptable for a function.

2. One-to-One (Injective):
Every input leads to a unique output. No two different inputs share the same output value.
Example: \(f(x) = 2x + 1\). If \(x_1 \neq x_2\), then \(f(x_1) \neq f(x_2)\).

The Horizontal Line Test (HLT)

The HLT helps us determine if a function is One-to-One.

• If you draw a horizontal line anywhere on the graph and it crosses the graph at more than one point, the function is NOT one-to-one (it is many-to-one).
• If a function passes both the VLT and the HLT, it is called a Bijective function.

Why is One-to-One important? A function must be one-to-one to have an inverse function (we'll cover that soon!).

2.3 Symmetry: Even and Odd Functions (HL Focus)

Symmetry allows us to predict the behavior of a function across the axes.

1. Even Functions (Symmetry about the Y-axis):

• Definition: \(f(-x) = f(x)\)
• If you substitute \(-x\) for \(x\), the function remains unchanged.
• Example: \(f(x) = x^2 + 5\).

2. Odd Functions (Symmetry about the Origin):

• Definition: \(f(-x) = -f(x)\)
• If you substitute \(-x\) for \(x\), the entire function becomes the negative of the original function.
• Example: \(f(x) = x^3 - x\).

Did you know? Most functions are neither even nor odd!

3. Function Transformations: Shifting, Stretching, and Reflecting

Transformations allow us to take a known, simple graph (the parent function) and move it, stretch it, or flip it without having to replot every point.

Let \(y = f(x)\) be the original function. We are looking at the effect of \(y = a f(b(x-h)) + k\).

🧠 The Transformation Trick (Mnemonic Aid)

Think of transformations as happening in two places:

• Outside the function (affecting \(y\)): Behave exactly as you expect.
• Inside the function (affecting \(x\)): Behave the opposite of what you expect.

3.1 Translations (Shifts)

Translations move the graph without changing its shape or orientation.

• Vertical Shift (Outside): \(y = f(x) + k\)
  If \(k > 0\), shift up by \(k\). If \(k < 0\), shift down by \(|k|\).
• Horizontal Shift (Inside): \(y = f(x - h)\)
  If \(h > 0\), shift right by \(h\). If \(h < 0\), shift left by \(|h|\). (Remember: opposite!)

3.2 Stretches and Compressions

Stretches change the shape of the graph, making it narrower or wider.

• Vertical Stretch/Compression (Outside): \(y = a \cdot f(x)\)
  Stretch factor \(a\). If \(|a| > 1\), it's a vertical stretch. If \(0 < |a| < 1\), it's a vertical compression.
• Horizontal Stretch/Compression (Inside): \(y = f(b x)\)
  Stretch factor \(1/b\). If \(|b| > 1\), it's a horizontal compression by factor \(1/b\). If \(0 < |b| < 1\), it's a horizontal stretch by factor \(1/b\). (Remember: opposite!)

3.3 Reflections

Reflections flip the graph over an axis.

• Reflection in the X-axis (Outside): \(y = -f(x)\)
  The graph flips vertically (all \(y\)-values switch signs).
• Reflection in the Y-axis (Inside): \(y = f(-x)\)
  The graph flips horizontally (all \(x\)-values switch signs).

4. Operations on Functions: Composites and Inverses

4.1 Composite Functions

A composite function is created when the output of one function becomes the input for another function. It's chaining two (or more) functions together.

Notation: \((f \circ g)(x)\) is read as "f composed with g of x," and means \(f(g(x))\).

Analogy: You are making coffee (\(g\)). The output is brewed coffee. You then put that brewed coffee into a frother (\(f\)). The output of \(g\) is the input for \(f\).

Step-by-Step Calculation:

1. Identify the inner function, \(g(x)\).
2. Substitute the entire expression for \(g(x)\) into the outer function, \(f\), wherever you see \(x\).

Example: If \(f(x) = x^2 + 1\) and \(g(x) = 2x\).
\((f \circ g)(x) = f(g(x)) = f(2x) = (2x)^2 + 1 = 4x^2 + 1\).

Domain of a Composite Function (HL Rigour):

The domain of \(f(g(x))\) is tricky! It must satisfy two conditions:

1. \(x\) must be in the domain of the inner function, \(g\).
2. The output \(g(x)\) must be in the domain of the outer function, \(f\).

4.2 Inverse Functions

The inverse function, denoted \(f^{-1}(x)\), undoes what the original function \(f(x)\) did.

Key Requirement: A function can only have an inverse if it is One-to-One (it must pass the HLT). If it fails the HLT (like \(f(x) = x^2\)), we must restrict the domain of the original function to force it to be one-to-one before finding the inverse.

Finding the Inverse Function (\(f^{-1}(x)\))

This is a mechanical, step-by-step process:

1. Replace \(f(x)\) with \(y\).
   Example: \(y = 3x - 5\)
2. Swap \(x\) and \(y\) (This is the algebraic step that defines the inverse relationship).
   Example: \(x = 3y - 5\)
3. Solve the new equation for \(y\).
   Example: \(x + 5 = 3y \implies y = \frac{x+5}{3}\)
4. Replace \(y\) with \(f^{-1}(x)\).
   Example: \(f^{-1}(x) = \frac{x+5}{3}\)

The Relationship Between Domains and Ranges:

The domain of \(f(x)\) becomes the range of \(f^{-1}(x)\).
The range of \(f(x)\) becomes the domain of \(f^{-1}(x)\).

This relationship is critical for defining the inverse correctly, especially when working with restricted domains.

Graphically, \(f(x)\) and \(f^{-1}(x)\) are reflections of each other across the line \(y = x\).

Did you know? The composition of a function and its inverse always equals the input: \[ (f \circ f^{-1})(x) = x \] \[ (f^{-1} \circ f)(x) = x \] If you get \(x\) when you compose them, you know you found the correct inverse!