Functions

Welcome to the World of Functions!

In this chapter, we are going to explore Functions, which are one of the most important building blocks of Algebra. Think of a function as a "mathematical machine": you put a number in (the input), the machine does something to it, and a specific number comes out (the output). Understanding how these machines work, how to combine them, and how to reverse them is key to mastering A Level Maths.

Don't worry if some of the notation looks like a different language at first. We will break everything down step-by-step, starting with the very basics!

1. The Language of Functions

To talk about functions like a pro, we need to use the right vocabulary. There are three main "ingredients" to every function:

1. The Mapping: This is the rule that tells us what to do with the input. We usually write it as \(f(x) = \dots\) or \(f : x \mapsto \dots\).
2. The Domain: This is the set of all possible input values (usually \(x\)) that you are allowed to put into the function.
3. The Range: This is the set of all possible output values (usually \(f(x)\) or \(y\)) that the function can produce.

Mappings: One-to-One vs. Many-to-One

For a rule to be officially called a Function, each input must have exactly one output. Imagine a vending machine: if you press the button for "Cola," you expect to get a Cola every time. If sometimes it gave you Cola and sometimes it gave you Water, the machine would be "broken"—it wouldn't be a function!

• One-to-One (1:1): Every unique input gives a unique output. Example: \(f(x) = x + 3\). If you put in different numbers, you always get different results.
• Many-to-One: Different inputs can result in the same output. Example: \(f(x) = x^2\). Both \(x = 2\) and \(x = -2\) give the output \(4\). This is still a function because each input still only has one specific output.

Quick Review: If an input has more than one output (One-to-Many), it is not a function. It's just a mapping.

Did you know? You can use the "Vertical Line Test" on a graph to see if it's a function. If you can draw a vertical line anywhere that hits the graph more than once, it's not a function!

Key Takeaway: A function is a rule where every input in the domain has exactly one output in the range. Functions can be one-to-one or many-to-one.

2. Composite Functions

A Composite Function is basically what happens when you "chain" two machines together. You take the output of the first function and use it as the input for the second function.

The notation looks like this: \(gf(x)\).
Important: In math, we work from the inside out. So, \(gf(x)\) means you apply function \(f\) first, then apply function \(g\) to that result.

Step-by-Step: How to find \(gf(x)\)

Let's say \(f(x) = 2x + 1\) and \(g(x) = x^2\).
1. Look at the outer function: \(g(\dots)\).
2. Replace every \(x\) in the outer function with the entire inner function expression.
3. So, \(gf(x) = (2x + 1)^2\).
4. Simplify if needed: \(gf(x) = 4x^2 + 4x + 1\).

Common Mistake: Doing them in the wrong order! \(gf(x)\) is usually very different from \(fg(x)\). Always start with the function closest to the \(x\).

Key Takeaway: \(gf(x)\) means "do \(f\) first, then do \(g\)." It's like a relay race where \(f\) passes the baton to \(g\).

3. Inverse Functions

An Inverse Function, written as \(f^{-1}(x)\), is the "undo" button. If the original function takes \(x\) to \(y\), the inverse takes \(y\) back to \(x\).

The Golden Rule for Inverses

A function must be One-to-One to have an inverse. Why? Because if a function is many-to-one (like \(x^2\)), the "undo" button wouldn't know which original value to go back to!

How to find the Inverse Algebraically

1. Set the function equal to \(y\): \(y = f(x)\).
2. Rearrange the equation to make \(x\) the subject.
3. Swap the \(x\) and \(y\).
4. Replace the new \(y\) with \(f^{-1}(x)\).

Example: Find the inverse of \(f(x) = 3x - 5\)
1. \(y = 3x - 5\)
2. \(y + 5 = 3x \implies x = \frac{y + 5}{3}\)
3. Swap: \(y = \frac{x + 5}{3}\)
4. \(f^{-1}(x) = \frac{x + 5}{3}\)

Graphing the Inverse

The graph of \(y = f^{-1}(x)\) is a reflection of the graph \(y = f(x)\) in the line \(y = x\). This is a very common exam question!

Key Takeaway: To have an inverse, a function must be one-to-one. To find it, rearrange to make \(x\) the subject and then swap the variables.

4. The Modulus Function

The Modulus of a number, written as \(|x|\), is its "absolute value." In simple terms: it makes everything positive!

• \(|5| = 5\)
• \(|-5| = 5\)

Graphs of \(y = |ax + b|\)

The graph of a linear modulus function always looks like a "V" shape. The "point" of the V (the vertex) happens where the stuff inside the bars equals zero.

Example: Sketch \(y = |x - 3|\)
1. If this were \(y = x - 3\), it would be a straight line crossing the x-axis at \(3\).
2. Because of the modulus, the part of the line that would go below the x-axis is reflected upwards.
3. The result is a "V" with the corner at \((3, 0)\).

Solving Modulus Equations

When you see \(|f(x)| = k\), it usually means there are two possibilities to solve:
1. \(f(x) = k\) (The "normal" version)
2. \(f(x) = -k\) (The "reflected" version)

Don't Panic: If you are solving an inequality like \(|x + 2| \leq |2x - 1|\), a great trick is to square both sides. Since both sides are positive, squaring them removes the modulus bars and leaves you with a quadratic to solve!

Key Takeaway: The modulus \(|x|\) turns negative values into positive ones, creating V-shaped graphs and requiring us to consider both positive and negative cases when solving.

5. Graph Transformations

This is all about how we can move or stretch a graph. We compare the "new" function to the "original" \(y = f(x)\).

Translations (Moving)

• \(y = f(x) + a\): Moves the graph up by \(a\) units. (Vertical translation)
• \(y = f(x + a)\): Moves the graph left by \(a\) units. (Horizontal translation)

Memory Trick: Changes outside the brackets affect the y-axis and are "normal" (+ is up). Changes inside the brackets affect the x-axis and are "weird/opposite" (+ is left)!

Stretches (Squashing and Pulling)

• \(y = a \cdot f(x)\): A vertical stretch by scale factor \(a\). (Multiplies all y-coordinates by \(a\)).
• \(y = f(ax)\): A horizontal stretch by scale factor \(\frac{1}{a}\). (Divides all x-coordinates by \(a\)).

Quick Example: If you have \(y = f(2x)\), you actually squash the graph horizontally so it's half as wide. If you have \(y = 2f(x)\), you pull it vertically so it's twice as tall.

Combining Transformations

When multiple things happen (e.g., \(y = 2f(x + 3)\)), the order matters.
• For Vertical changes (outside): Follow the order of operations (BIDMAS). Multiply/Stretch first, then Add/Translate.
• For Horizontal changes (inside): It's usually easier to think about the translation first, but be careful! Stick to one change at a time.

Common Mistake: Forgetting that horizontal stretches use \(\frac{1}{a}\). If you see \(f(3x)\), the scale factor is \(\frac{1}{3}\), not \(3\)!

Key Takeaway: Outside changes are vertical and "normal." Inside changes are horizontal and "opposite." Use these rules to shift and stretch your basic graphs into new positions.

Final Summary Review

• Functions take one input to exactly one output.
• Domain = inputs; Range = outputs.
• Composite \(gf(x)\) means do \(f\) then \(g\).
• Inverse \(f^{-1}(x)\) reflects the graph in \(y=x\) and only exists for 1:1 functions.
• Modulus \(|x|\) makes everything positive and creates "V" graphs.
• Transformations move graphs: \(f(x+a)\) is horizontal (left/right), \(f(x)+a\) is vertical (up/down).

You've got this! Practice sketching these and identifying domains and ranges, and you'll be a Functions expert in no time!