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Composition of Functions & Domains Worksheet | Advanced PreCalculus

Composition of Functions & Domains Worksheet | Advanced PreCalculus

Subject: Mathematics

Age range: 14-16

Resource type: Worksheet/Activity

Last updated

11 August 2026

pdf, 104.85 KB
pdf, 104.85 KB

A worksheet on composing functions. It includes harder exercises on this topic where the domain of the composition has to be determined based on the domains of the two original functions. Detailed solutions are included.

Master composite functions and finding domains with this rigorous, ready-to-print pure maths worksheet! Perfect for A-Level Maths, KS5, and IB Mathematics students, this resource includes fully worked step-by-step solutions to save you valuable marking time.

Challenge your advanced secondary and college maths students with this no-prep composite functions and domains practice worksheet! This resource is specifically engineered to move past basic substitution and push students toward true conceptual mastery of composite algebraic relationships and inner/outer domain analysis.
With 8 multi-step, advanced problems, this printable activity targets the exact rigor required for upper-level secondary maths and sixth-form college courses. Students will face complex compositions involving surds, algebraic fractions, restricted interval boundaries, and working backward to find missing components.

What’s Included in This Resource?
• 8 High-Level Practice Problems: Questions requiring advanced algebraic manipulation to find composite formulas and their resulting definitions.
• Focus on Complex Restrictions: Multiple problems dedicated entirely to evaluating step-by-step domain constraints (including intersections of square roots, algebraic fractions, and limited piece-like intervals).
• Decomposing Functions Practice: A targeted concluding challenge forcing students to work backward from a final composite output to solve for an unknown inner function.
• Fully Worked Solutions: Step-by-step mathematical answers for every single problem to facilitate effortless marking or student self-checking.

Core Skills Covered:
• Finding composite equations for f of g of x, g of f of x, and self-compositions like f of f of x.
• Determining strict domain boundaries for compositions involving surds (radicals) and denominators (rational numbers).
• Evaluating composite domain solutions using interval notation and algebraic inequality sets.
• Decomposing complex functions to solve for an original inner variable rule.

Perfect For:
• AS & A-Level Mathematics (Pure Maths specifications)
• Key Stage 5 (KS5) / Sixth Form College
• GCSE Further / Additional Mathematics Extension
• International Baccalaureate (IB) Maths Higher Level (HL)
• Rigorous Homework, Exam Revision, Extension Work, or Cover Plans

Exam Board Alignment:
• Excellent preparation for Edexcel, AQA, OCR, and Cambridge International (CIE) specifications covering functions, domains, ranges, and composite operations.

Get this resource as part of a bundle and save up to 54%

A bundle is a package of resources grouped together to teach a particular topic, or a series of lessons, in one place.

Bundle

PreCalculus Functions Bundle | Composite, Inverse, Even Odd Worksheet Pack

A collection of 8 worksheets on functions. The following topics are included: * Identifying Graphs of Functions * Identifying Functions, Domain and Range-Set of Ordered Pairs * Evaluating Functions * Even and Odd Functions * Equality of Functions * Composition of Functions * Injective Functions * Inverse Functions (harder) Solutions are provided. Master advanced function mechanics with this complete A-Level Maths worksheet bundle! Perfectly optimized for Year 12 and Year 13 Pure Mathematics revision, this print-and-go pack covers composite functions, inverse proofs, odd/even functions, and piecewise evaluation with full step-by-step worked solutions. Looking for a rigorous, print-and-go resource pack to help your sixth-form students confidently master advanced functions? This Advanced Functions Mastery Worksheet Bundle contains a highly structured collection of problems designed to build fluency and deep analytical confidence. Moving from foundational index and mapping rules to tricky rational composition domains and formal inverse function derivations, this resource is perfect for daily secondary lessons, independent revision booklets, structured homework assignments, cover work, or university preparation! Every question includes an uncrowded, student-friendly layout and comes accompanied by a complete, step-by-step worked answer key making marking quick and straightforward for busy teachers or peer assessment. **What's Included in This Bundle?** • **Even and Odd Functions:** Fluency practice using algebraic substitutions and graphical charts to classify function symmetry. • **Equality of Functions:** Analyzing subset conditions to establish where two algebraic relationships match. • **Composition of Functions:** Extensive practice calculating composite functions alongside tracking rigorous domain conditions. • **Injective Functions:** Applying formal one-to-one definitions and horizontal graph checks. • **Inverse Functions:** Proving invertibility and rearranging equations to isolate the inverse function for rational, radical, and piecewise curves. • **Evaluating Functions:** Traditional calculation practice including nested parameters and multi-step piecewise functions. **Perfect For:** • AS & A-Level Mathematics: Targeted Pure Maths practice for Year 12 and Year 13 students pushing for maximum marks (Grades A and A*). • Advanced Higher / IB Diploma: Rigorous algebraic training for upper-level secondary curriculum paths. • Cover Work & Extensions: Self-contained layouts requiring zero prep work from you or a supply teacher.

£7.95
Bundle

Function Operations, Composition & Equality Bundle Algebra 2 PreCalculus

A collection of three worksheets on the following topics: * Function Operations: adding, subtracting, multiplying and dividing functions and determining the domain of the functions produced * Equality of Functions: examining whether two given functions are equal * Composition of Functions: composing functions including harder cases on this topic where the domain of the composition has to be determined based on the domains of the two original functions Detailed solutions are included. Master advanced function relationships with this comprehensive, no-prep functions worksheet bundle! Perfect for A-Level Maths, KS5, and IB Mathematics classrooms, this 3-resource collection features advanced domain analysis and fully worked step-by-step solutions to save you valuable marking time. ________________________________________ Help your secondary and sixth-form college maths students master advanced algebraic mappings with this highly structured, print-and-go Function Operations, Composition, and Equality Worksheet Bundle! This cost-saving collection contains three distinct, rigorously designed resources engineered to take students beyond basic arithmetic calculations. Instead, it challenges them with deep conceptual skills like evaluating intersecting domain constraints, analyzing identical function behaviours, working with absolute values (modulus functions), and performing multi-step algebraic operations. Whether you need targeted homework assignments, rigorous revision lessons, no-prep cover plans, or complete exam revision, this comprehensive bundle provides the exact practice your upper-level algebra students need. **What’s Included in This Bundle?** This premium bundle includes three complete worksheets, each packed with a detailed, step-by-step mathematical answer key: **1. Equality of Functions Worksheet** o Analyzing Functional Pairs: Students examine pairs of rational, radical, and modulus functions to determine if they are mathematically equal. o Domain Constraint Optimization: For functions that are not equal, students must determine the largest possible subset of real numbers where the functions coincide (e.g., handling square roots of squares or factoring out algebraic fractions). **2. Composition of Functions Worksheet** o Rigorous Compositions: Covers multi-step algebraic manipulations to find composite functions like fg(x), gf(x), and self-compositions like ff(x). o Advanced Restrictions: Includes complex nested functions involving fractional denominators, surds (square roots), and defined interval sets. o Decomposing Functions Challenge: A targeted section forcing students to work backward from a final composite output to solve for an unknown inner function rule. **3. Function Operations Worksheet** o Arithmetic Combinations: Students solve for the sum, difference, product, and quotient of given expressions. o Determining Domain Intersections: Highlights finding precisely where combined functions are undefined or limited by overlapping surd bounds. **Core Skills Covered:** • Performing addition, subtraction, multiplication, and division on multiple function types. • Finding composite equations and stating domain restrictions using interval notation. • Formulating backwards solutions to decompose a nested algebraic variable. • Evaluating function domains to prove equivalence or identify restricted shared subsets. • Working confidently with rational algebraic fractions, surds, and modulus functions. **Perfect For:** • AS & A-Level Mathematics (Pure Maths specifications) • Key Stage 5 (KS5) / Sixth Form College • GCSE Further / Additional Mathematics Extension • International Baccalaureate (IB) Maths Higher Level (HL) • Rigorous Homework, Exam Revision, Extension Work, or Cover Plans **Exam Board Alignment:** • Excellent preparation for Edexcel, AQA, OCR, and Cambridge International (CIE) specifications covering functions, domains, ranges, operations, and composite mappings. ________________________________________

£3.50

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