In this section, we delve into the concepts of range and composition in functions, which are fundamental topics in CIE A-Level Maths. Grasping these concepts is essential for analysing and applying functions effectively in various mathematical scenarios.

**Finding the Range of a Function**

The range of a function refers to the set of all possible output values (y-values) it can produce, given its domain (input values). Identifying the range is key to understanding the breadth of a function's output.

**Steps to Find the Range**

**Determine Extremes**: Identify the highest and lowest y-values possible, based on the function's domain.**Quadratic Functions**: For functions like $f(x) = 3x^2 + 5x - 6$, use the method of completing the square to find the vertex, which helps determine the range.**Vertex as Minimum**: If the coefficient of $x^2$ is positive, the vertex indicates the minimum point of the function.**Vertex as Maximum**: If the coefficient of $x^2$ is negative, the vertex indicates the maximum point of the function.

**Composition of Two Functions**

Function composition involves creating a new function by applying one function to the results of another, effectively combining two functions into a single operation.

**Definition and Notation**

**Composition Notation**: Composition of two functions $f$ and $g$ is denoted as $(f \circ g)(x)$, read as "f composed with g of x."**Operational Aspect**: The composition $(f \circ g)(x)$ is defined as $f(g(x))$, meaning the output of $(g(x)$ becomes the input for $f(x)$.

**Examples**

**Basic Composition**:

- Let $f(x) = 4x + 5$ and $g(x) = x^2 - 5$.
- Then, $(f \circ g)(x) = f(g(x)) = 4(g(x)) + 5 = 4(x^2 - 5) + 5$.

**Inverse Functions**:

- Suppose $f(x) = \sqrt{x}$ and $g(x) = x^2$, where $x \geq 0$.
- The composition $(f \circ g)(x) = f(g(x)) = \sqrt{x^2} = |x|$, but since $x \geq 0 ), ( (f \circ g)(x) = x .$

**Trigonometric Functions**:- Consider $f(x) = \sin(x)$ and $g(x) = \cos(x)$.
- The composition $(f \circ g)(x) = f(g(x)) = \sin(\cos(x))$yields a new function combining sine and cosine.

**Domain and Range Considerations**

**Compatibility**: A composite function like $(f \circ g)(x)$ can only be formed when the range of $g(x)$ is within the domain of $f(x)$.**Analysis**: To ensure the composition is valid, the domain and range of each function must be analysed before composing them.

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.