Pure Mathematics | Algebra and functions

Welcome to Algebra and Functions!

Hello future mathematicians! This chapter is the absolute foundation of your journey in Pure Mathematics 1. Think of algebra as your crucial toolbox—the skills you learn here (like handling powers, roots, and rearranging equations) will be used in every single topic, from geometry to calculus.

Don't worry if some of these rules seem abstract right now. We'll break them down using simple analogies and step-by-step instructions. By the end of this section, you will be fluent in manipulating indices, simplifying surds, and understanding basic function notation!

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Section 1: Mastering Indices (Powers)

Indices (or powers) tell you how many times a number is multiplied by itself. Understanding the laws of indices is essential for simplifying complex expressions quickly.

1.1 The Laws of Indices

These laws work for any base \(a\) and any indices \(m\) and \(n\).

• Law 1: Multiplication: When multiplying powers with the same base, you add the indices.
  Example: \(a^m \times a^n = a^{m+n}\)
  (e.g., \(x^3 \times x^4 = x^{3+4} = x^7\))


• Law 2: Division: When dividing powers with the same base, you subtract the indices.
  Example: \(a^m \div a^n = a^{m-n}\)
  (e.g., \(y^6 \div y^2 = y^{6-2} = y^4\))


• Law 3: Power of a Power: When raising a power to another power, you multiply the indices.
  Example: \((a^m)^n = a^{mn}\)
  (e.g., \((2^3)^5 = 2^{15}\))

1.2 Special Index Rules

The Zero Power

Any non-zero number raised to the power of zero is always 1.

Rule: \(a^0 = 1\) (where \(a \neq 0\)).
(e.g., \(5^0 = 1\), \((x^2y)^0 = 1\))

The Negative Power

A negative index means you take the reciprocal (flip the fraction). This is a common trap for students!

Rule: \(a^{-n} = \frac{1}{a^n}\) and \(\frac{1}{a^{-n}} = a^n\)
(e.g., \(4^{-2} = \frac{1}{4^2} = \frac{1}{16}\))

Memory Aid: A negative index signals that the term is in the "wrong" place (numerator or denominator), so you must flip it to make the power positive.

Fractional Powers (Roots)

Fractional powers relate directly to roots. The denominator of the fraction tells you the type of root.

Rule 1 (Unit Fraction): \(a^{1/n} = \sqrt[n]{a}\)
(e.g., \(8^{1/3} = \sqrt[3]{8} = 2\))

Rule 2 (General Fraction): \(a^{m/n} = (\sqrt[n]{a})^m\) or \(\sqrt[n]{a^m}\)
(e.g., \(25^{3/2} = (\sqrt{25})^3 = 5^3 = 125\))

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Section 2: Surds (Irrational Roots)

A surd is a root (like a square root or cube root) that results in an irrational number—a number that cannot be written exactly as a fraction and has non-repeating, non-terminating decimals (e.g., \(\sqrt{2}\) or \(\sqrt{7}\)).

We keep numbers as surds to maintain exact answers, which is a key requirement in P1 mathematics.

2.1 Simplifying Surds

To simplify a surd, look for the largest square number that is a factor of the number inside the root.

Rule: \(\sqrt{ab} = \sqrt{a} \times \sqrt{b}\)

Step-by-Step Example: Simplifying \(\sqrt{72}\)

1. Find the largest perfect square factor of 72. (Perfect squares are 4, 9, 16, 25, 36...)
2. We see that 36 is a factor: \(72 = 36 \times 2\).
3. Separate the surd: \(\sqrt{72} = \sqrt{36} \times \sqrt{2}\)
4. Simplify the perfect square: \(\sqrt{72} = 6\sqrt{2}\).

2.2 Rationalising the Denominator

In maths, it is considered bad form to leave a surd in the denominator of a fraction. Rationalising means changing the denominator into an integer.

Case 1: Single Surd in the Denominator

If the denominator is \(\sqrt{a}\), multiply the numerator and denominator by \(\sqrt{a}\).

Example: Rationalise \(\frac{3}{\sqrt{5}}\)
\[ \frac{3}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{3\sqrt{5}}{5} \]

Case 2: Denominator is \(a \pm \sqrt{b}\) (Using the Conjugate)

If the denominator has two terms (one or both are surds, like \(4 + \sqrt{3}\)), you must multiply by the conjugate.

• The conjugate of \(a + \sqrt{b}\) is \(a - \sqrt{b}\).
• The conjugate of \(a - \sqrt{b}\) is \(a + \sqrt{b}\).

Multiplying by the conjugate uses the Difference of Two Squares rule: \((x+y)(x-y) = x^2 - y^2\). This eliminates the surd!

Step-by-Step Example: Rationalise \(\frac{1}{4 - \sqrt{3}}\)

1. Identify the conjugate: The conjugate of \(4 - \sqrt{3}\) is \(4 + \sqrt{3}\).
2. Multiply top and bottom by the conjugate:
   \[ \frac{1}{4 - \sqrt{3}} \times \frac{4 + \sqrt{3}}{4 + \sqrt{3}} \]
3. Simplify the numerator: \(1 \times (4 + \sqrt{3}) = 4 + \sqrt{3}\).
4. Simplify the denominator (using \(a^2 - b^2\)): \((4 - \sqrt{3})(4 + \sqrt{3}) = 4^2 - (\sqrt{3})^2 = 16 - 3 = 13\).
5. The result is: \(\frac{4 + \sqrt{3}}{13}\).

Key Takeaway (Surds): Always provide answers in their simplest, exact form, and never leave a surd in the denominator.

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Section 3: Algebraic Expressions and Fractions

P1 requires fluency in expanding, factorising, and manipulating algebraic fractions, particularly those involving polynomials.

3.1 Expanding and Factorising

Expanding involves multiplying out brackets (using methods like FOIL for two brackets).
Example: \((x + 3)(2x - 1) = 2x^2 - x + 6x - 3 = 2x^2 + 5x - 3\)

Factorising is the reverse process—putting an expression back into brackets. For P1, you must be confident with:

• Common Factors: \(4x^2 - 6x = 2x(2x - 3)\)
• Difference of Two Squares (DOTS): \(a^2 - b^2 = (a-b)(a+b)\)
  (e.g., \(9x^2 - 25 = (3x - 5)(3x + 5)\))
• Factorising Quadratics (This is covered more deeply in the next chapter, but skills like finding two numbers that multiply to C and add to B are essential).

3.2 Algebraic Fractions

Working with algebraic fractions follows the exact same rules as working with regular numerical fractions.

Simplifying Fractions

To simplify, factorise the numerator and denominator and look for common factors to cancel out.

Example: Simplify \(\frac{x^2 + 3x}{x^2 - 9}\)

1. Factorise numerator (common factor): \(x(x + 3)\)
2. Factorise denominator (DOTS): \((x - 3)(x + 3)\)
3. The fraction becomes: \(\frac{x(x + 3)}{(x - 3)(x + 3)}\)
4. Cancel the common factor \((x + 3)\). The result is: \(\frac{x}{x - 3}\).

Common Mistake to Avoid: You can only cancel common factors, not common terms. You cannot cancel the \(x^2\) in \(\frac{x^2 + 2}{x^2 + 5}\) because they are terms within an addition/subtraction, not factors of the whole expression.

Adding and Subtracting Fractions

You need a Common Denominator (LCD).

Step-by-Step Example: \(\frac{2}{x+1} + \frac{3}{x}\)

1. The LCD is the product of the two denominators: \(x(x+1)\).
2. Adjust the first fraction: \(\frac{2}{x+1} \times \frac{x}{x} = \frac{2x}{x(x+1)}\)
3. Adjust the second fraction: \(\frac{3}{x} \times \frac{x+1}{x+1} = \frac{3(x+1)}{x(x+1)}\)
4. Combine the numerators:
   \[ \frac{2x + 3(x+1)}{x(x+1)} = \frac{2x + 3x + 3}{x(x+1)} = \frac{5x + 3}{x^2 + x} \]

Did you know? Algebra was developed independently by ancient Babylonians, Greeks, and Indians, but the term 'Algebra' comes from the Arabic word 'al-jabr', meaning 'the reunion of broken parts'.

Key Takeaway (Expressions): Always factorise first when simplifying complex fractions. Look out for DOTS!

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Section 4: Introducing Functions (Notation, Domain, and Range)

A function is a rule that maps every input value (x) to exactly one output value (y or f(x)).

4.1 Function Notation

Instead of writing \(y = 3x + 2\), we often use function notation:

\[ f(x) = 3x + 2 \]

This is read as "f of x equals 3x plus 2." It means that \(f\) is the name of the function, and \(x\) is the input variable.

Evaluating Functions

To evaluate \(f(a)\), substitute \(a\) for every \(x\) in the function rule.

Example: If \(f(x) = x^2 - 5\), find \(f(4)\).
\[ f(4) = (4)^2 - 5 = 16 - 5 = 11 \] Example: Find \(f(2a)\).
\[ f(2a) = (2a)^2 - 5 = 4a^2 - 5 \]

4.2 Domain and Range

The concepts of domain and range are crucial because they define the boundaries of the function.

The Domain (Input)

The Domain is the complete set of all possible input values (\(x\)) for which the function is defined.

• For most simple polynomials (e.g., \(f(x) = x^2 + 5x - 1\)), the domain is all real numbers (\(x \in \mathbb{R}\)).
• Restrictions: In P1, the domain is often restricted if:
  1. The function involves division (cannot divide by zero).
  2. The function involves square roots (cannot square root a negative number).

Example Restriction: For \(g(x) = \frac{1}{x-3}\), we cannot have \(x-3=0\). Therefore, the domain is \(x \neq 3\).

The Range (Output)

The Range is the complete set of all possible output values (\(f(x)\) or \(y\)) that the function can produce.

• The range is often harder to determine and usually requires sketching the graph or using completing the square (for quadratics, covered later).
• For simple functions like \(f(x) = x^2\), since squaring any real number results in a number \(\ge 0\), the range is \(f(x) \ge 0\).

4.3 Notation for Inequalities and Sets

In P1, you must use correct mathematical notation for domains and ranges:

• "x is greater than 5": \(x > 5\)
• "y is less than or equal to 10": \(y \le 10\)
• "x is between 2 and 7 (inclusive)": \(2 \le x \le 7\)
• "x belongs to the set of real numbers": \(x \in \mathbb{R}\)

Key Takeaway (Functions): \(f(x)\) is the output associated with the input \(x\). The domain defines what \(x\) values are allowed; the range defines what \(f(x)\) values result.