Algebra III (AQA GCSE Further Maths): Revision Notes

Graphs of functions with up to three parts to their domains

What are piecewise functions?

A piecewise function is a special type of function that has different rules or expressions for different parts of its domain. Think of it like having different instructions depending on which range of x-values you're working with. These functions are incredibly useful for modelling real-world situations where behaviour changes at certain points.

When we write a piecewise function, we specify:

• The expression or rule for each part
• The domain (range of x-values) where each rule applies
• Whether endpoints are included ($\leq$, $\geq$) or excluded ($\<$, $>$)

Here's what makes piecewise functions special: the complete domain is the union of all the individual parts, and crucially, no x-value should appear in more than one part of the domain. This ensures the function gives exactly one output for each input.

Understanding domain notation

When working with piecewise functions, you'll encounter inequality symbols that tell you exactly where each rule applies:

• $\leq$ or $\geq$: The endpoint is included (shown as a filled circle on graphs)
• $\<$ or $>$: The endpoint is excluded (shown as an empty circle on graphs)

The key rule to remember is that each x-value must belong to exactly one part of the domain - no overlaps allowed!

Step-by-step graphing approach

Let's work through how to graph piecewise functions systematically using a detailed example.

The result is a function that changes behaviour at the boundary points, creating a unique graph that combines different mathematical shapes.

Reading graphs to find function definitions

Sometimes you'll be given a graph and asked to write the piecewise function definition. Here's how to approach this systematically:

Real-world applications: velocity and distance

Piecewise functions are particularly useful for modelling motion problems. Let's examine a practical example involving a car's journey.

In velocity-time problems, the graph shows how speed changes over time. Different phases of motion (acceleration, constant speed, deceleration) create natural segments in the piecewise function.

For rectangular or trapezoidal shapes under the curve, you can calculate areas using geometric formulas:

• Rectangle: $\text{length} \times \text{width}$
• Trapezium: $\frac{1}{2} \times (\text{sum of parallel sides}) \times \text{height}$

Common function types in piecewise definitions

When working with piecewise functions, you'll frequently encounter these types:

Constant functions: $f(x) = k$

• Create horizontal line segments
• Gradient is zero
• Useful for representing periods of no change

Linear functions: $f(x) = mx + c$

• Create straight line segments
• Constant gradient $m$
• Can be increasing ($m > 0$) or decreasing ($m \< 0$)

Quadratic functions: $f(x) = ax^2 + bx + c$

• Create curved parabolic segments
• Can open upward ($a > 0$) or downward ($a \< 0$)
• Often used for sections involving acceleration or deceleration

Exam tips and common pitfalls