Algebra and functions

Welcome to Algebra and Functions!

Welcome to one of the most important chapters in your A Level Mathematics journey. Think of Algebra and Functions as the "toolbox" for the rest of the course. Whether you are solving complex calculus problems or modeling the path of a projectile in Mechanics, the skills you learn here will be your best friends.

Don’t worry if some of this feels like a big step up from GCSE—we’re going to break it down into small, bite-sized pieces. By the end of these notes, you’ll be handling polynomials and transforming graphs like a pro!

1. Power Up: Laws of Indices and Surds

Before we dive into functions, we need to make sure our basic algebra is rock solid. Indices (powers) and surds (roots) follow very specific rules.

The Laws of Indices

You’ll remember these from before, but now we use them with rational exponents (fractions).
1. Multiplication: \( a^m \times a^n = a^{m+n} \)
2. Division: \( a^m \div a^n = a^{m-n} \)
3. Power of a Power: \( (a^m)^n = a^{mn} \)
4. Fractional Indices: \( a^{\frac{m}{n}} = \sqrt[n]{a^m} \) or \( (\sqrt[n]{a})^m \)

Memory Aid: When the bases are multiplying, the powers are adding. Think of it like a team getting stronger!

Surds and Rationalising

A surd is just an irrational root. To "rationalise the denominator" means to get rid of any square roots on the bottom of a fraction.
To rationalise \( \frac{1}{\sqrt{x} + \sqrt{y}} \), we multiply the top and bottom by the conjugate: \( (\sqrt{x} - \sqrt{y}) \).
Example: \( \frac{1}{\sqrt{3} + 2} \) becomes \( \frac{1(\sqrt{3} - 2)}{(\sqrt{3} + 2)(\sqrt{3} - 2)} \).
The bottom simplifies using the difference of two squares: \( (\sqrt{3})^2 - (2)^2 = 3 - 4 = -1 \).

Quick Review:
• \( \sqrt{xy} = \sqrt{x}\sqrt{y} \)
• Always check if a surd can be simplified (e.g., \( \sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5} \)).

2. Quadratic Functions: The "U" Shape

Quadratics are functions in the form \( f(x) = ax^2 + bx + c \). They create a curve called a parabola.

The Discriminant: The Root Detector

The discriminant is the part of the quadratic formula under the square root: \( b^2 - 4ac \). It tells you how many times the graph hits the x-axis:
• If \( b^2 - 4ac > 0 \): Two distinct real roots (The graph crosses the x-axis twice).
• If \( b^2 - 4ac = 0 \): One repeated real root (The graph just touches the x-axis).
• If \( b^2 - 4ac < 0 \): No real roots (The graph is floating above or below the x-axis).

Completing the Square

This is a superpower for finding the vertex (the turning point) of a graph.
The form is: \( a(x + \frac{b}{2a})^2 + (c - \frac{b^2}{4a}) \).
The turning point is simply \( (-\frac{b}{2a}, \text{the constant at the end}) \).

Common Mistake: Forgetting to divide \( b \) by 2, or forgetting to square the fraction when subtracting it at the end!

Did you know? Quadratics can be "hidden." You might see \( 2^{2x} + 2^x - 6 = 0 \). If you let \( u = 2^x \), it becomes a simple quadratic: \( u^2 + u - 6 = 0 \)!

3. Simultaneous Equations and Inequalities

When solving equations with two variables (like \( x \) and \( y \)), you have two main methods: substitution and elimination. For A Level, if one equation is linear and the other is quadratic, substitution is usually your only option.

Quadratic Inequalities

Solving \( x^2 - 5x + 6 < 0 \) isn't as simple as linear equations.
1. Find the critical values by solving the equation as if it were \( = 0 \). (Here, \( x=2, x=3 \)).
2. Sketch the graph.
3. If it’s \( < 0 \), you want the part below the x-axis (one interval: \( 2 < x < 3 \)).
4. If it’s \( > 0 \), you want the parts above the x-axis (two separate intervals: \( x < 2 \) or \( x > 3 \)).

Key Takeaway: Always sketch the curve for inequalities! It prevents you from guessing the wrong direction for the "arrows."

4. Polynomials and the Factor Theorem

A polynomial is just an expression with many terms, like a cubic (\( x^3 \)) or a quartic (\( x^4 \)).

The Factor Theorem

This is a huge time-saver! If you plug a number \( a \) into a function and get 0 (i.e., \( f(a) = 0 \)), then \( (x - a) \) is a factor of that polynomial.
Conversely, if you are told \( (ax - b) \) is a factor, then \( f(\frac{b}{a}) = 0 \).

Algebraic Division

When you divide a polynomial by \( (ax + b) \), you can simplify complex fractions or find other factors. It’s just like long division with numbers—just take it one term at a time!

5. Graphs and Transformations

You need to recognize the shapes of cubics, quartics, and reciprocal graphs (\( y = \frac{a}{x} \) and \( y = \frac{a}{x^2} \)).

Asymptotes

An asymptote is a line that the graph gets closer and closer to but never actually touches.
For \( y = \frac{2}{x+a} + b \):
• The vertical asymptote is \( x = -a \) (because you can't divide by zero!).
• The horizontal asymptote is \( y = b \).

Modulus Functions

The modulus sign \( |x| \) means "the positive version of." It acts like a mirror.
To sketch \( y = |f(x)| \), draw the original graph and any part that is below the x-axis gets reflected upwards to become positive. It often creates a sharp "V" shape.

Graph Transformations

This is where students often get confused, so here is a simple trick:
• Outside the brackets: Affects \( y \). Does exactly what it says.
Example: \( f(x) + a \) moves the graph up by \( a \).
• Inside the brackets: Affects \( x \). Does the opposite of what it says.
Example: \( f(x + a) \) moves the graph left by \( a \) (even though it's a plus!).

Memory Aid: "In is x-tra weird." (Inside the brackets affects \( x \), and it's weird because it's the opposite).

6. Composite and Inverse Functions

A function is like a machine: you put an input (\( x \)) in, and it gives you one output (\( y \)).

• Domain: The set of possible inputs (the x-values).
• Range: The set of possible outputs (the y-values).

Composite Functions: \( fg(x) \)

This means "put \( x \) into \( g \), then take that answer and put it into \( f \)."
Always work from the inside out!

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

The inverse function "undoes" the original.
• To find it: Replace \( f(x) \) with \( y \), swap \( x \) and \( y \), and rearrange to make \( y \) the subject.
• The Graph: The graph of \( f^{-1}(x) \) is a reflection of \( f(x) \) in the line \( y = x \).

7. Partial Fractions

Sometimes we need to break a large fraction into smaller, simpler ones. This is called decomposition.

• Linear Factors: \( \frac{1}{(x+a)(x+b)} = \frac{A}{x+a} + \frac{B}{x+b} \)
• Repeated Factors: \( \frac{1}{(x+a)(x+b)^2} = \frac{A}{x+a} + \frac{B}{x+b} + \frac{C}{(x+b)^2} \)

Don't forget that extra term for the repeated factor!

Key Takeaway for Partial Fractions: Multiply through by the denominator and pick smart values for \( x \) (like the roots) to find \( A, B, \) and \( C \) quickly.

Summary Checklist

Can you...
• Rationalise a denominator using the conjugate?
• Use the discriminant to check for roots?
• Solve a quadratic inequality using a sketch?
• Use the Factor Theorem to find roots of a cubic?
• Sketch modulus graphs and transformations?
• Find an inverse function by swapping \( x \) and \( y \)?
• Split a rational expression into partial fractions?

If you can do these, you've mastered the core of Algebra and Functions! Keep practicing, and don't be afraid to sketch graphs—they are your visual map to the answer.