Which domain of the function between f and g is equal to domain gof(x)

Chapter 1

Foundations: domain, range, and restrictions

1

Core definitions and notation (domain, range/image, codomain)

2

Common domain restrictions (denominators, square roots, even roots, logs)

3

Range/image of a function (why it matters for composition)

2 subtopics

4

Finding range from algebra and from graphs (basic techniques)

5

Ranges of common “parent” functions (linear, quadratic, reciprocal, sqrt, exp, log, trig basics)

Chapter 2

Function composition basics

6

Definition: (g ∘ f)(x) = g(f(x)) and what it means

7

Order matters: comparing g ∘ f vs f ∘ g

8

Composing using tables and graphs (when you don’t have formulas)

Chapter 3

Domain of the composite function g ∘ f

9

Key rule: Dom(g ∘ f) = { x ∈ Dom(f) | f(x) ∈ Dom(g) }

10

Step-by-step procedure to compute Dom(g ∘ f)

3 subtopics

11

Step 1: start with the domain of f (possible inputs to the composite)

12

Step 2: enforce that f(x) lies in Dom(g) (output of f must be legal input to g)

13

Step 3: intersect conditions and write the final domain (set-builder / interval notation)

↗ Range/image of a function (why it matters for composition) (see Chapter 1)

14

When does Dom(g ∘ f) equal Dom(f)? When is it a strict subset?

Chapter 4

Worked examples (increasing difficulty)

15

Algebraic examples (formulas given)

2 subtopics

16

Example: compositions involving radicals/logs (extra restrictions come from g)

17

Example: compositions involving rational functions (denominator restrictions after substitution)

18

Piecewise-function examples (domain depends on which piece f(x) falls into)

1 subtopics

19

Example: piecewise f with a restricted-domain g (compute Dom(g ∘ f) carefully)

20

Graph-based examples (domain/range read from graphs)

1 subtopics

21

Example: using graph of f to ensure f(x) lands inside Dom(g)

Chapter 5

Practice and self-check

22

Quick-check questions (conceptual: subset relations, equality conditions)

23

Mixed practice set: find Dom(g ∘ f) for 10–15 pairs (with answer key)

24

Common mistakes (assuming Dom(g ∘ f)=Dom(g), forgetting substitution changes restrictions)