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

1. Determine Extremes: Identify the highest and lowest y-values possible, based on the function's domain.
2. 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

1. Basic Composition:

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

1. 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 $x \geq 0 ), ( (f \circ g)(x) = x .$

1. 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.