Welcome to the World of Functions!
In this chapter, we are going to explore one of the most important concepts in Mathematics: Functions. Think of a function as a "mathematical machine." You put a number in (the input), the machine does some work, and a specific number comes out (the output). Understanding functions is like learning the rules of the game for almost everything else you will do in Pure Mathematics 1.
We will cover how to describe these machines, how to link them together, how to reverse them, and how to change the way they look on a graph. Don't worry if some of this feels a bit abstract at first—we'll break it down piece by piece!
1. What Exactly is a Function?
A function is a special relationship where every single input has exactly one output. If you put the same number into the machine twice, you must get the same result both times.
Key Terms:
• Domain: The set of all possible "input" values (the \( x \) values). Think of this as what the machine is allowed to eat.
• Range: The set of all possible "output" values (the \( y \) or \( f(x) \) values). This is what the machine produces.
Analogy: Imagine a soda machine. If you press the "Cola" button (input), you expect a Cola (output). If you press "Cola" and sometimes get Ginger Ale, that machine is broken—it's not a function!
How to find the Range
In your exams, you'll often be asked to find the range for a given domain.
Example: If \( f(x) = x^2 \) for the domain \( x \geq 1 \).
If you plug in the smallest value (\( x = 1 \)), you get \( 1^2 = 1 \). Since \( x \) gets bigger from there, the outputs will also get bigger. So, the Range is \( f(x) \geq 1 \).
Quick Review: To find the range, look at the graph or plug the boundaries of the domain into the function. Watch out for turning points (like the vertex of a quadratic)!
2. One-One Functions
A one-one function is a very strict type of function. Not only does every input have one output, but every output comes from exactly one input. No two different inputs can ever give the same output.
The Horizontal Line Test: If you draw a horizontal line anywhere on a graph and it hits the curve more than once, it is not one-one.
Example: \( f(x) = x^2 \) for all real numbers is not one-one because both \( x = 2 \) and \( x = -2 \) give the output \( 4 \). However, if we restrict the domain to \( x \geq 0 \), it becomes one-one!
3. Composite Functions
Composition is what happens when you link two machines together. The output of the first machine becomes the input for the second.
Notation: \( gf(x) \) means "apply \( f \) first, then apply \( g \) to the result."
Memory Aid: Always work from right to left. In \( gf(x) \), \( f \) is closest to \( x \), so it goes first.
The Golden Rule for Composition:
You can only form the composite function \( gf \) if the Range of \( f \) fits inside the Domain of \( g \). If the first machine produces something the second machine isn't allowed to eat, the whole system crashes!
Example: If \( f(x) = x + 1 \) and \( g(x) = x^2 \), then \( gf(x) = g(x + 1) = (x + 1)^2 \).
4. Inverse Functions
An inverse function, written as \( f^{-1}(x) \), is a machine that runs in reverse. It takes the output and brings you back to the original input.
Important Note: Only one-one functions have inverses. If a function isn't one-one, the "reverse machine" wouldn't know which input to go back to!
How to find the Inverse (Step-by-Step):
1. Write the function as \( y = ... \)
2. Rearrange the equation to make \( x \) the subject.
3. Swap \( x \) and \( y \).
4. Replace \( y \) with \( f^{-1}(x) \).
Example: Find the inverse of \( f(x) = 2x + 3 \).
• \( y = 2x + 3 \)
• \( y - 3 = 2x \)
• \( x = \frac{y - 3}{2} \)
• \( f^{-1}(x) = \frac{x - 3}{2} \)
The Graph Connection
The graph of \( y = f^{-1}(x) \) is a reflection of the graph \( y = f(x) \) in the line \( y = x \). This is because we are simply swapping the \( x \) and \( y \) coordinates.
Did you know? The Domain of \( f \) is exactly the same as the Range of \( f^{-1} \), and the Range of \( f \) is the Domain of \( f^{-1} \). They swap everything!
5. Graph Transformations
Sometimes we want to move or stretch a graph. There are four main types you need to know for Paper 1.
Translations (Shifts)
• Vertical Shift: \( y = f(x) + a \)
Moves the graph up by \( a \) units. (If \( a \) is negative, it moves down).
• Horizontal Shift: \( y = f(x + a) \)
Moves the graph left by \( a \) units.
Warning: This one is tricky! Adding \( a \) moves it in the negative direction (left), and subtracting \( a \) moves it right. It's the opposite of what you might expect!
Stretches
• Vertical Stretch: \( y = a \cdot f(x) \)
Stretches the graph vertically by scale factor \( a \). The \( y \)-coordinates are multiplied by \( a \).
• Horizontal Stretch: \( y = f(ax) \)
Stretches the graph horizontally by scale factor \( \frac{1}{a} \).
Warning: Just like the shift, the horizontal change is "inside out." If you see \( 2x \), the graph actually gets narrower (factor of \( 1/2 \)).
Reflections
• \( y = -f(x) \): Reflection in the x-axis (upside down).
• \( y = f(-x) \): Reflection in the y-axis (left-to-right swap).
Takeaway Table:
• \( f(x) + a \): Translation \( \begin{pmatrix} 0 \\ a \end{pmatrix} \)
• \( f(x + a) \): Translation \( \begin{pmatrix} -a \\ 0 \end{pmatrix} \)
• \( a \cdot f(x) \): Vertical Stretch, factor \( a \)
• \( f(ax) \): Horizontal Stretch, factor \( \frac{1}{a} \)
Common Mistakes to Avoid
• Order of Operations: When doing multiple transformations, the order matters! Usually, work from the "inside" of the brackets outward.
• Domain Restrictions: Always check if the question gives a specific domain. This affects your range and whether an inverse exists.
• Notation: Don't confuse \( f^{-1}(x) \) with \( \frac{1}{f(x)} \). They are completely different things! One is the inverse, the other is the reciprocal.
• Range of Quadratics: To find the range of a quadratic, you must find the vertex (turning point) by completing the square. The highest or lowest point is crucial for the range.
Quick Review Summary
• A function maps one input to exactly one output.
• Domain = Inputs; Range = Outputs.
• Composite \( gf(x) \) means do \( f \) first, then \( g \).
• Inverse \( f^{-1} \) only exists for one-one functions; reflect in \( y = x \).
• Horizontal changes (inside the brackets) always feel "backwards" or "opposite."
Don't worry if this seems tricky at first! Practice sketching these transformations and finding inverses, and soon it will feel like second nature. You've got this!