International Mathematics (0607) | Functions

Welcome to the World of Functions!

Hello future mathematicians! This chapter is all about Functions—one of the most fundamental and powerful concepts in advanced mathematics. Don't worry if the notation looks strange at first; functions are simply rules that link inputs to outputs.

Think of a function as a sophisticated machine: you put something in (the input), the machine follows a rule, and something specific comes out (the output).

Mastering functions is crucial because they allow us to model real-world relationships—from calculating compound interest to predicting the path of a rocket! Grab your Graphic Display Calculator (GDC); it's going to be your best friend in this unit!

Section 1: Understanding Functions and Notation (C3.3 / E3.3)

1.1 What is a Function?

A function is a relationship where every input has exactly one output.

We use function notation to describe this rule. Instead of writing \(y = 2x + 1\), we write:
\(f(x) = 2x + 1\)

Key Terminology:

• \(f(x)\): This is read as "f of x," and it represents the output (which is usually the \(y\)-value).
• Input (\(x\)): The value you put into the function.
• Rule (\(f\)): The operation applied to the input.

Step-by-Step: Using Function Notation

If \(f(x) = 3x - 5\), finding \(f(2)\) means replacing every \(x\) with 2:
1. Write the function: \(f(x) = 3x - 5\)
2. Substitute \(x=2\): \(f(2) = 3(2) - 5\)
3. Calculate: \(f(2) = 6 - 5 = 1\)
So, when the input is 2, the output is 1.

1.2 Domain and Range (Extended Only - E3.3)

The domain and range tell us exactly what values a function can handle.

The Domain (Inputs):

• The set of all possible input values (\(x\)) for which the function is defined.
• Analogy: If your function is a machine that crushes rocks, the domain might be "rocks up to 10kg." You can't put in a feather or a 20kg boulder!

The Range (Outputs):

• The set of all possible output values (\(y\)) that the function produces.
• Analogy: If your machine turns rocks into dust, the range is "dust" (and not, say, "liquid water").

Section 2: Recognizing Graphs of Functions (C3.1 / E3.1)

Being able to identify the type of function just by looking at its graph is a vital skill.

2.1 Core Functions (C3.1)

1. Linear Function
Formula: \(f(x) = ax + b\)
Shape: A straight line.
(The coefficient \(a\) is the gradient, and \(b\) is the y-intercept.)

2. Quadratic Function
Formula: \(f(x) = ax^2 + bx + c\)
Shape: A parabola (U-shape or inverted U-shape).
If \(a > 0\), it's a happy face (opens upwards). If \(a < 0\), it's a sad face (opens downwards).

2.2 Extended Functions (E3.1 - Recognizing shapes and features)

3. Cubic Function
Formula: \(f(x) = ax^3 + bx^2 + cx + d\)
Shape: An S-shape, often with one local maximum and one local minimum (turning points). It extends from top-right to bottom-left (or vice versa).

4. Reciprocal Function
Formula: \(f(x) = \frac{k}{x}\) (where \(k\) is a constant)
Shape: Has two separate parts (branches) and approaches asymptotes (lines it never touches, see E3.5).

5. Exponential Function
Formula: \(f(x) = a^x\) (where \(a > 0\))
Shape: Rises or falls very quickly. It has a horizontal asymptote (usually the \(x\)-axis).
If \(a > 1\), the function grows (exponential growth). If \(0 < a < 1\), the function decays (exponential decay).

6. Trigonometric Functions
Formulas: \(f(x) = a \sin(bx)\), \(f(x) = a \cos(bx)\), \(f(x) = \tan x\)
Shape: Periodic (repeats itself).

• Sine and Cosine: Smooth, wave-like curves that oscillate between a maximum and minimum value.
• Tangent: Repeats every 180° and has vertical asymptotes (e.g., at 90°, 270°, etc.).

The value \(a\) controls the amplitude (height of the wave).
The value \(b\) controls the period (how quickly it repeats).

Section 3: Using the Graphic Display Calculator (GDC) (C3.2 / E3.2)

Your GDC is essential for solving function problems without tedious manual algebra, especially for non-linear or unfamiliar functions.

3.1 Essential GDC Skills Checklist

You must be able to use your GDC to:

1. Sketch a graph: Enter the function and display the visual representation.
2. Produce a table of values: Useful for checking points or plotting manually.
3. Find Zeros (or Roots): Find the \(x\)-intercepts, i.e., where \(f(x) = 0\).
4. Find Local Maxima or Local Minima: Find the turning points of non-linear graphs (like quadratics or cubics).
5. Find the Intersection Point: Find the coordinates where two graphs meet.
6. Find the Vertex of a Quadratic: The special name for the minimum or maximum turning point of a parabola.

Common Mistake to Avoid: When finding intersection points or zeros, always look at the specific range requested in the question. The GDC will find all points, but you only need the ones in the required domain!

Section 4: Advanced Function Operations (Extended Only - E3.3)

4.1 Inverse Functions (\(f^{-1}(x)\))

The inverse function reverses the original function. If \(f\) takes \(x\) to \(y\), then \(f^{-1}\) takes \(y\) back to \(x\).

Graphically, the graph of \(y = f^{-1}(x)\) is a reflection of \(y = f(x)\) across the line \(y = x\).

Step-by-Step: Finding the Inverse Function

Let \(f(x) = 5x + 8\).
1. Replace \(f(x)\) with \(y\):
\(y = 5x + 8\)
2. Swap \(x\) and \(y\) (this reverses the relationship):
\(x = 5y + 8\)
3. Rearrange to make \(y\) the subject:
\(x - 8 = 5y\)
\(y = \frac{x - 8}{5}\)
4. Write using inverse notation:
\(f^{-1}(x) = \frac{x - 8}{5}\)

Memory Aid: To find the inverse, think of the three S's: Substitute \(y\), Swap \(x\) and \(y\), and Solve for \(y\).

4.2 Composite Functions

A composite function is created when you apply one function immediately after another.

If we have two functions, \(f(x)\) and \(g(x)\), we can combine them:

• \(fg(x)\): Means you apply \(g\) first, then apply \(f\) to the result.
• \(gf(x)\): Means you apply \(f\) first, then apply \(g\) to the result.

Step-by-Step: Forming Composite Functions

Let \(f(x) = x + 3\) and \(g(x) = 2x^2\). Find \(gf(x)\).
1. Start with the outer function, \(g\). Replace the input of \(g\) with the inner function, \(f(x)\).
\(g(f(x)) = 2(f(x))^2\)
2. Substitute the expression for \(f(x)\) into the formula:
\(gf(x) = 2(x + 3)^2\)
3. Expand (if required by the question):
\(gf(x) = 2(x^2 + 6x + 9) = 2x^2 + 12x + 18\)

Important Note (E3.3): You are not expected to find the domain and range of composite functions, only to form them and express them in their simplest form.

Section 5: Advanced Graph Features and Transformations (Extended Only)

5.1 Finding a Quadratic Function (E3.4)

When given specific points, you can determine the equation of a quadratic function, which is often written in the vertex form:
\(y = a(x - h)^2 + k\)

The benefit of this form is that the vertex (the turning point) is immediately visible at the coordinates \((h, k)\).

• If given the vertex \((h, k)\) and another point:
  1. Substitute \((h, k)\) into \(y = a(x - h)^2 + k\).
  2. Substitute the coordinates of the other point \((x, y)\) into the equation.
  3. Solve for the value of \(a\).
• If given the \(x\)-intercepts (\(p\) and \(q\)) and another point:
  You can also use the intercept form: \(y = a(x - p)(x - q)\).

5.2 Asymptotes (E3.5)

An asymptote is a line that a graph approaches closer and closer to, but never actually touches. They act as invisible boundary lines.

You need to identify simple asymptotes that are parallel to the axes.

• In the reciprocal function \(y = \frac{1}{x}\), the graph has two asymptotes:
  - Vertical Asymptote: \(x = 0\) (The \(y\)-axis)
  - Horizontal Asymptote: \(y = 0\) (The \(x\)-axis)
• In the trigonometric function \(f(x) = \tan x\), the graph has vertical asymptotes at 90°, 270°, etc.

5.3 Transforming Graphs (Translations) (E3.6)

Transformations describe how the graph of \(y = f(x)\) changes when you modify the function. In this syllabus, we focus on translations (shifting the graph).

Let \(k\) be a positive integer (a constant value).

1. Vertical Translation (Shift Up or Down)
Formula: \(y = f(x) + k\)
Effect: Shifts the graph vertically up by \(k\) units.
Formula: \(y = f(x) - k\)
Effect: Shifts the graph vertically down by \(k\) units.

2. Horizontal Translation (Shift Left or Right)
Formula: \(y = f(x + k)\)
Effect: Shifts the graph horizontally left by \(k\) units.
Formula: \(y = f(x - k)\)
Effect: Shifts the graph horizontally right by \(k\) units.

Memory Trick: Vertical translations are intuitive (\(+k\) moves up). Horizontal translations are counter-intuitive or "Lies" (\(x + k\) moves left, which feels like the negative direction!).

5.4 The Logarithmic Function (E3.7)

The logarithmic function is the inverse of the exponential function.

If we have an exponential relationship:
\(y = a^x\)
This is equivalent to the logarithmic relationship:
\(x = \log_a y\)
(Read as: "x is the logarithm of y to the base a.")

In IGCSE 0607, all logarithms are assumed to be base 10 unless stated otherwise. We usually write this as \(\log y\).

Solving Equations using Logs

If you need to solve an exponential equation where the unknown is in the power, logarithms are the tool to use.

To solve \(a^x = b\), we convert it to the log form:
\(x = \frac{\log b}{\log a}\)

Example: Solve \(5^x = 100\).
\(x = \frac{\log 100}{\log 5}\)
\(x \approx \frac{2}{0.699}\) (You use your calculator for the final value).