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CIE A-Level Maths Study Notes

1.2.3 Range and Composition

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)=3x2+5x6f(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 x2x^2 is positive, the vertex indicates the minimum point of the function.
    • Vertex as Maximum: If the coefficient of x2x^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 ff and gg is denoted as (fg)(x)(f \circ g)(x), read as "f composed with g of x."
  • Operational Aspect: The composition (fg)(x)(f \circ g)(x) is defined as f(g(x))f(g(x)), meaning the output of (g(x)(g(x) becomes the input for f(x)f(x).


  1. Basic Composition:
  • Let f(x)=4x+5f(x) = 4x + 5 and g(x)=x25g(x) = x^2 - 5.
  • Then, (fg)(x)=f(g(x))=4(g(x))+5=4(x25)+5(f \circ g)(x) = f(g(x)) = 4(g(x)) + 5 = 4(x^2 - 5) + 5.
  1. Inverse Functions:
  • Suppose f(x)=xf(x) = \sqrt{x} and g(x)=x2g(x) = x^2, where x0x \geq 0.
  • The composition (fg)(x)=f(g(x))=x2=x(f \circ g)(x) = f(g(x)) = \sqrt{x^2} = |x|, but since x0),((fg)(x)=x.x \geq 0 ), ( (f \circ g)(x) = x .
  1. Trigonometric Functions:
    • Consider f(x)=sin(x)f(x) = \sin(x) and g(x)=cos(x)g(x) = \cos(x).
    • The composition (fg)(x)=f(g(x))=sin(cos(x))(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 (fg)(x)(f \circ g)(x) can only be formed when the range of g(x)g(x) is within the domain of f(x)f(x).
  • Analysis: To ensure the composition is valid, the domain and range of each function must be analysed before composing them.

Practice Questions



Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
Oxford University - PhD Mathematics

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.

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