AMP STUDY NOTES
MAT1512 · CALCULUS A
Complete Module
Study Notes
All five chapters’ study notes combined into a single volume, in
order, for the full module.
Chapter 1 — Functions and Models
Chapter 2 — Limits and Derivatives
Chapter 3 — Differentiation Rules
Chapter 4 — Integrals
Chapter 5 — Differential Equations, Growth/Decay & Partial Derivatives
STUDY NOTES · ALL CHAPTERS
Independent study material. Independently authored revision aid, not affiliated with, endorsed by, or sourced
from UNISA or any official assessment body. © AMP Study Notes.
,AMP STUDY NOTES
MAT1512 · CALCULUS A
Chapter 1
Functions and Models
Exam-focused study notes: domain and range, piecewise-defined
functions, even and odd symmetry, composite functions, and inverse
functions — with fully worked practice problems and complete step-
by-step solutions.
STUDY NOTES · CHAPTER 1
Independent study material. These notes are an independently authored revision aid based on the standard
first-year calculus syllabus (following the treatment in Stewart, Calculus: Early Transcendentals) and on general
patterns observed in how this material tends to be assessed. They are not affiliated with, endorsed by, or sourced
from the University of South Africa (UNISA) or any official assessment body, and all practice problems are original
variations written for study purposes, with independently worked solutions. © AMP Study Notes.
, AMP Study Notes — MAT1512 — Chapter 1
Chapter 1: Functions and Models
TOPICS COVERED IN THIS CHAPTER
• Functions, domain, and range • Combining functions: composite functions and their
domains
• Piecewise-defined functions
• Inverse functions, and the domain/range swap
• Symmetry: even and odd functions
• A quick survey of standard function "models":
polynomial, exponential, logarithmic, trigonometric
MAT1512 Calculus A (Unisa). This chapter is foundational rather than heavily examined on its own
— it establishes the vocabulary and algebraic manipulation skills (domain-finding, composing,
inverting) that every later chapter leans on constantly, especially once implicit differentiation and the
chain rule are in play. Based on the module's own tutorial letter and a review of recent exam sittings,
the recurring high-yield tasks are: finding the domain of a function built from roots, fractions, and logs
in combination; forming a composite function and correctly determining its domain (which can be
more restrictive than either original function's own domain); and finding the inverse of a one-to-one
function algebraically. Determining whether a function is even, odd, or neither also appears as a quick,
self-contained question type.
What is Most Examined?
HIGH-YIELD TASK HOW IT TENDS TO APPEAR PRIORITY
Domain of a combined Find the domain of an expression combining a square root, a Very high
function denominator, and/or a logarithm — each restriction must be found and
intersected.
Composite functions Form or , simplify, and find the domain of the High
and their domain composite, not just of or alone.
Inverse functions Solve for in terms of , then swap variable names to get High
; verify by composition.
Even/odd classification Test against and algebraically. Medium
Piecewise functions Evaluate at a point, or sketch/interpret, a function defined by different Medium
formulas on different intervals.
Page 2 of 7
, AMP Study Notes — MAT1512 — Chapter 1
Table of Contents
1. Functions, Domain, and Range
2. Piecewise Functions and Symmetry
3. Composite Functions
4. Inverse Functions
5. Worked Examples
6. Common Mistakes
7. Exam Checklist
8. Key Equations Summary
Functions, Domain, and Range
A function assigns to every input in a set called the domain exactly one output ; the set of all outputs
actually produced is the range. When a function is given only by a formula (no domain stated explicitly), the
convention is to take the domain to be every real for which the formula actually produces a real number — this is
the natural domain, and finding it is the single most common opening task in this chapter.
THREE DOMAIN RESTRICTIONS TO ALWAYS CHECK
• Denominators: exclude any that makes a denominator zero.
• Even roots ( ): the expression under the root must be .
• Logarithms: the argument of or must be (strictly — zero is not allowed either).
When a formula has more than one of these features at once, find each restriction separately, then take the
intersection of all of them — not the union. A value has to be simultaneously safe for every part of the formula to
belong to the domain.
e.g. needs ; if it also had a factor, would additionally be excluded, giving domain
.
Piecewise Functions and Symmetry
A piecewise-defined function uses a different formula on different parts of its domain (a common example being
itself, defined as for and for ). To evaluate one at a specific point, first identify which interval
that point falls into, then use only the formula attached to that interval.
EVEN AND ODD FUNCTIONS
is even if for every in the domain — its graph is a mirror image across the -axis. is
odd if — its graph has rotational symmetry about the origin. Many functions are
neither.
Page 3 of 7
MAT1512 · CALCULUS A
Complete Module
Study Notes
All five chapters’ study notes combined into a single volume, in
order, for the full module.
Chapter 1 — Functions and Models
Chapter 2 — Limits and Derivatives
Chapter 3 — Differentiation Rules
Chapter 4 — Integrals
Chapter 5 — Differential Equations, Growth/Decay & Partial Derivatives
STUDY NOTES · ALL CHAPTERS
Independent study material. Independently authored revision aid, not affiliated with, endorsed by, or sourced
from UNISA or any official assessment body. © AMP Study Notes.
,AMP STUDY NOTES
MAT1512 · CALCULUS A
Chapter 1
Functions and Models
Exam-focused study notes: domain and range, piecewise-defined
functions, even and odd symmetry, composite functions, and inverse
functions — with fully worked practice problems and complete step-
by-step solutions.
STUDY NOTES · CHAPTER 1
Independent study material. These notes are an independently authored revision aid based on the standard
first-year calculus syllabus (following the treatment in Stewart, Calculus: Early Transcendentals) and on general
patterns observed in how this material tends to be assessed. They are not affiliated with, endorsed by, or sourced
from the University of South Africa (UNISA) or any official assessment body, and all practice problems are original
variations written for study purposes, with independently worked solutions. © AMP Study Notes.
, AMP Study Notes — MAT1512 — Chapter 1
Chapter 1: Functions and Models
TOPICS COVERED IN THIS CHAPTER
• Functions, domain, and range • Combining functions: composite functions and their
domains
• Piecewise-defined functions
• Inverse functions, and the domain/range swap
• Symmetry: even and odd functions
• A quick survey of standard function "models":
polynomial, exponential, logarithmic, trigonometric
MAT1512 Calculus A (Unisa). This chapter is foundational rather than heavily examined on its own
— it establishes the vocabulary and algebraic manipulation skills (domain-finding, composing,
inverting) that every later chapter leans on constantly, especially once implicit differentiation and the
chain rule are in play. Based on the module's own tutorial letter and a review of recent exam sittings,
the recurring high-yield tasks are: finding the domain of a function built from roots, fractions, and logs
in combination; forming a composite function and correctly determining its domain (which can be
more restrictive than either original function's own domain); and finding the inverse of a one-to-one
function algebraically. Determining whether a function is even, odd, or neither also appears as a quick,
self-contained question type.
What is Most Examined?
HIGH-YIELD TASK HOW IT TENDS TO APPEAR PRIORITY
Domain of a combined Find the domain of an expression combining a square root, a Very high
function denominator, and/or a logarithm — each restriction must be found and
intersected.
Composite functions Form or , simplify, and find the domain of the High
and their domain composite, not just of or alone.
Inverse functions Solve for in terms of , then swap variable names to get High
; verify by composition.
Even/odd classification Test against and algebraically. Medium
Piecewise functions Evaluate at a point, or sketch/interpret, a function defined by different Medium
formulas on different intervals.
Page 2 of 7
, AMP Study Notes — MAT1512 — Chapter 1
Table of Contents
1. Functions, Domain, and Range
2. Piecewise Functions and Symmetry
3. Composite Functions
4. Inverse Functions
5. Worked Examples
6. Common Mistakes
7. Exam Checklist
8. Key Equations Summary
Functions, Domain, and Range
A function assigns to every input in a set called the domain exactly one output ; the set of all outputs
actually produced is the range. When a function is given only by a formula (no domain stated explicitly), the
convention is to take the domain to be every real for which the formula actually produces a real number — this is
the natural domain, and finding it is the single most common opening task in this chapter.
THREE DOMAIN RESTRICTIONS TO ALWAYS CHECK
• Denominators: exclude any that makes a denominator zero.
• Even roots ( ): the expression under the root must be .
• Logarithms: the argument of or must be (strictly — zero is not allowed either).
When a formula has more than one of these features at once, find each restriction separately, then take the
intersection of all of them — not the union. A value has to be simultaneously safe for every part of the formula to
belong to the domain.
e.g. needs ; if it also had a factor, would additionally be excluded, giving domain
.
Piecewise Functions and Symmetry
A piecewise-defined function uses a different formula on different parts of its domain (a common example being
itself, defined as for and for ). To evaluate one at a specific point, first identify which interval
that point falls into, then use only the formula attached to that interval.
EVEN AND ODD FUNCTIONS
is even if for every in the domain — its graph is a mirror image across the -axis. is
odd if — its graph has rotational symmetry about the origin. Many functions are
neither.
Page 3 of 7
