IGCSE Mathematics (0606) – Complete Syllabus Notes
1. Functions – definitions, domain, range, composite & inverse
- Function. A rule that assigns to each element $x$ in a set $X$ exactly one element $y$ in a set $Y$. Notation: $y=f(x)$ or $f:x\mapsto y$.
- Domain. Set of all admissible inputs. Write $\operatorname{Dom}(f)$.
- Range (image). Set of all actual outputs $y=f(x)$. Write $\operatorname{Ran}(f)$.
- Codomain. The set $Y$ that the function is declared to map into. In most IGCSE questions $Y=\mathbb R$ unless stated otherwise.
- Composite function. $(f\circ g)(x)=f\bigl(g(x)\bigr)$. Order matters: $f\circ geq g\circ f$ in general.
- Inverse function. $f^{-1}$ satisfies $f^{-1}(f(x))=x$ and $f(f^{-1}(x))=x$ for every $x$ in the appropriate domain. Exists only when $f$ is one‑to‑one (bijective) on its domain.
1.1 Determining the domain
- Denominators – $x$ cannot make any denominator zero.
- Even roots – radicand $\ge0$.
- Logarithms – argument $>0$.
- Trigonometric restrictions – e.g. $\tan x$ undefined at $x=\frac{\pi}{2}+k\pi$.
- Composite – apply the above to each inner function before the outer one.
1.2 Determining the range
- Algebraic method. Set $y=f(x)$, solve for $x$ in terms of $y$, then apply the domain restrictions to $y$.
- Graphical reasoning. Sketch the curve (or recall the standard shape) and read off the $y$‑values attained.
- Known elementary ranges. $\sin x\in[-1,1]$, $e^{x}>0$, $\sqrt{x}\ge0$, etc.
- Codomain reminder. The range cannot exceed the declared codomain.
1.3 Absolute‑value transformation $y=|f(x)|$
The graph of $|f(x)|$ is obtained by reflecting any part of $y=f(x)$ that lies below the $x$‑axis upwards.
1.4 Inverses – existence & construction
- Check the horizontal‑line test (one‑to‑one).
- If not one‑to‑one, restrict the domain to a region where it is.
- Write $y=f(x)$, swap $x$ and $y$, solve for $y$ → $f^{-1}(x)$.
- State: domain$(f^{-1})=$ range$(f)$, range$(f^{-1})=$ domain$(f)$.
1.5 Composite functions – domain
$$\operatorname{Dom}(f\circ g)=\{\,x\in\operatorname{Dom}(g)\mid g(x)\in\operatorname{Dom}(f)\,\}.$$
1.6 Quick‑check checklist (functions)
| Task | Key steps | Typical answer form |
|---|
| Domain | List all restrictions, intersect them. | Interval or set notation. |
| Range | Solve $y=f(x)$ for $x$, apply domain. | Interval or set notation. |
| Composite $f\circ g$ | Compute $f(g(x))$, then apply the domain rule. | Domain set, expression for $f\circ g$. |
| Inverse | Horizontal‑line test → restrict if needed → swap & solve. | $f^{-1}(x)$ with domain & range. |
2. Quadratic Functions
- Standard form. $f(x)=ax^{2}+bx+c\;(aeq0)$.
- Vertex form. $f(x)=a(x-h)^{2}+k$, where $(h,k)$ is the vertex.
- Completing the square. $ax^{2}+bx+c=a\bigl[(x+\tfrac{b}{2a})^{2}-\tfrac{b^{2}-4ac}{4a^{2}}\bigr]$.
- Discriminant. $\Delta=b^{2}-4ac$ determines the nature of the roots:
- $\Delta>0$ – two distinct real roots, graph crosses $x$‑axis twice.
- $\Delta=0$ – one repeated real root, vertex touches the $x$‑axis.
- $\Delta<0$ – no real roots, graph does not meet the $x$‑axis.
- Axis of symmetry. $x=h=-\dfrac{b}{2a}$.
- Range. If $a>0$, $\operatorname{Ran}= [k,\infty)$; if $a<0$, $\operatorname{Ran}=(-\infty,k]$.
Example – vertex and range
Find the vertex and range of $f(x)=2x^{2}-8x+3$.
- Complete the square: $f(x)=2\bigl[x^{2}-4x\bigr]+3=2\bigl[(x-2)^{2}-4\bigr]+3=2(x-2)^{2}-5$.
- Vertex $(h,k)=(2,-5)$. Since $a=2>0$, range $[-5,\infty)$.
Quadratic inequalities
Solve $x^{2}-5x+6\le0$.
- Factor: $(x-2)(x-3)\le0$.
- Sign chart → solution $2\le x\le3$.
3. Polynomials (degree $\ge2$)
- Degree. Highest power of $x$ with a non‑zero coefficient.
- Remainder theorem. For $f(x)$ divided by $(x-a)$, remainder $=f(a)$.
- Factor theorem. $(x-a)$ is a factor of $f(x)$ iff $f(a)=0$.
- Long division / synthetic division. Used to factor higher‑degree polynomials.
- Behaviour at infinity. Sign of leading coefficient determines end‑behaviour.
Worked example – factorising a cubic
Factor $f(x)=x^{3}-6x^{2}+11x-6$.
- Test possible integer roots $a=\pm1,\pm2,\pm3,\pm6$. $f(1)=0$, so $(x-1)$ is a factor.
- Divide $f(x)$ by $(x-1)$ → $x^{2}-5x+6$.
- Factor quadratic: $(x-2)(x-3)$. Hence $f(x)=(x-1)(x-2)(x-3)$.
4. Solving Equations & Inequalities
4.1 Linear equations
Standard form $ax+b=c$. Solution $x=\dfrac{c-b}{a}$ (provided $aeq0$).
4.2 Rational equations
Clear denominators, solve the resulting polynomial, then check for extraneous roots (values that make any original denominator zero).
4.3 Absolute‑value equations
Two cases: $\;|A|=B\;(B\ge0)\;\Longrightarrow\;A=B\text{ or }A=-B$.
4.4 Graphical solution method
Plot $y=f(x)$ and $y=g(x)$; intersections give solutions of $f(x)=g(x)$. Useful for equations that are difficult to solve algebraically.
Example – solving $|2x-5|=x+1$
- Require $x+1\ge0\;\Rightarrow\;x\ge-1$.
- Case 1: $2x-5=x+1\;\Rightarrow\;x=6$ (satisfies $x\ge-1$).
- Case 2: $-(2x-5)=x+1\;\Rightarrow\;-2x+5=x+1\;\Rightarrow\;3x=4\;\Rightarrow\;x=\tfrac{4}{3}$ (also $\ge-1$).
- Solutions $x=6,\;\dfrac{4}{3}$.
5. Simultaneous Equations
- Linear – two variables. Methods: substitution, elimination, or matrix (Cramer's rule – optional).
- Non‑linear – e.g. quadratic & linear. Substitute the linear expression into the non‑linear one, solve, then back‑substitute.
Example – non‑linear pair
Solve $\begin{cases}y=2x+3\\x^{2}+y=7\end{cases}$.
- Substitute $y$: $x^{2}+2x+3=7\;\Rightarrow\;x^{2}+2x-4=0$.
- Quadratic formula: $x=\dfrac{-2\pm\sqrt{4+16}}{2}=\dfrac{-2\pm\sqrt{20}}{2}=-1\pm\sqrt5$.
- Find $y$ using $y=2x+3$ → $y=2(-1\pm\sqrt5)+3=1\pm2\sqrt5$.
6. Logarithmic & Exponential Functions
- Exponential form. $y=a^{x}$, $a>0$, $aeq1$.
- Logarithmic form. $y=\log_{a}x\iff a^{y}=x$.
- Key laws.
- $a^{m}a^{n}=a^{m+n}$
- $\bigl(a^{m}\bigr)^{n}=a^{mn}$
- $\log_{a}(mn)=\log_{a}m+\log_{a}n$
- $\log_{a}\!\left(\frac{m}{n}\right)=\log_{a}m-\log_{a}n$
- $\log_{a}m^{n}=n\log_{a}m$
- Change of base: $\displaystyle\log_{a}b=\frac{\log_{c}b}{\log_{c}a}$ (commonly $c=10$ or $e$).
- Graphs. Exponential: horizontal asymptote $y=0$, passes through $(0,1)$. Logarithmic: vertical asymptote $x=0$, passes through $(1,0)$. They are inverses.
- Solving equations. Isolate the exponential or logarithmic term, then apply the inverse function.
Example – solve $3^{2x-1}=27$
Write $27=3^{3}$, so $3^{2x-1}=3^{3}\Rightarrow2x-1=3\Rightarrow x=2$.
Example – solve $\log_{2}(x+3)=4$
Convert to exponential: $2^{4}=x+3\Rightarrow x=13$.
7. Straight‑Line Graphs
- Gradient–intercept form. $y=mx+c$, where $m$ is the gradient and $c$ the $y$‑intercept.
- Two‑point form. $m=\dfrac{y_{2}-y_{1}}{x_{2}-x_{1}}$.
- Parallel lines. Same gradient $m$.
- Perpendicular lines. Gradients satisfy $m_{1}m_{2}=-1$ (provided neither is vertical).
- Intercepts. $x$‑intercept at $x=-\dfrac{c}{m}$ (if $meq0$); $y$‑intercept $c$.
Worked example
Find the equation of the line passing through $(2,5)$ and parallel to $y=3x-4$.
- Parallel ⇒ same gradient $m=3$.
- Use point‑slope: $y-5=3(x-2)\;\Rightarrow\;y=3x-1$.
8. Circles
- Standard equation. $(x-h)^{2}+(y-k)^{2}=r^{2}$, centre $(h,k)$, radius $r$.
- General form. $x^{2}+y^{2}+Dx+Ey+F=0$ (complete the square to find centre & radius).
- Chord & diameter. If a chord is $2r\sin\theta$, where $\theta$ is the subtended angle at the centre.
- Tangent. A line touching the circle at exactly one point; condition: distance from centre to line equals $r$.
Example – find centre & radius
Given $x^{2}+y^{2}-6x+4y+9=0$, rewrite:
- $(x^{2}-6x)+(y^{2}+4y)=-9$
- $(x-3)^{2}-9+(y+2)^{2}-4=-9\;\Rightarrow\;(x-3)^{2}+(y+2)^{2}=4$.
- Centre $(3,-2)$, radius $r=2$.
9. Circular Measure
- Full circle $=360^{\circ}=2\pi\text{ rad}$.
- Conversion: $1\text{ rad}= \dfrac{180}{\pi}^{\circ}$, $1^{\circ}= \dfrac{\pi}{180}\text{ rad}$.
- Arc length. $s=r\theta$ (θ in radians).
- Sector area. $A=\dfrac{1}{2}r^{2}\theta$ (θ in radians).
Example – sector area
Find the area of a sector with radius $5\,$cm and central angle $45^{\circ}$.
- Convert: $\theta=45^{\circ}\times\frac{\pi}{180}=\frac{\pi}{4}$ rad.
- $A=\frac12\cdot5^{2}\cdot\frac{\pi}{4}= \frac{25\pi}{8}\approx9.82\text{ cm}^{2}$.
10. Trigonometry
- Definitions (right‑angled triangle). $\sin\theta=\dfrac{\text{opp}}{\text{hyp}}$, $\cos\theta=\dfrac{\text{adj}}{\text{hyp}}$, $\tan\theta=\dfrac{\text{opp}}{\text{adj}}$.
- Reciprocal functions. $\csc\theta=\dfrac1{\sin\theta}$, $\sec\theta=\dfrac1{\cos\theta}$, $\cot\theta=\dfrac1{\tan\theta}$.
- Pythagorean identities. $\sin^{2}\theta+\cos^{2}\theta=1$, $1+\tan^{2}\theta=\sec^{2}\theta$, $1+\cot^{2}\theta=\csc^{2}\theta$.
- Solving triangles. Use SOH‑CAH‑TOA, the sine rule, cosine rule, and area formula $\frac12ab\sin C$.
- Graphs of $\sin x$, $\cos x$, $\tan x$. Amplitude, period $2\pi$ (or $\pi$ for $\tan$), phase shift, vertical shift.
Example – solve $\sin\theta=0.6$ for $0^{\circ}\le\theta<360^{\circ}$
Principal value $\theta_{1}= \sin^{-1}0.6\approx36.9^{\circ}$. Second solution $\theta_{2}=180^{\circ}-\theta_{1}\approx143.1^{\circ}$. Hence $\theta=36.9^{\circ},\;143.1^{\circ}$.
11. Permutations, Combinations & Probability
- Factorial. $n! = n\,(n-1)\dots 1$, with $0!=1$.
- Permutation. $P(n,r)=\dfrac{n!}{(n-r)!}$ – ordered selection of $r$ objects from $n$.
- Combination. $C(n,r)=\binom{n}{r}= \dfrac{n!}{r!(n-r)!}$ – unordered selection.
- Probability (classical). $P(E)=\dfrac{\text{favourable outcomes}}{\text{total outcomes}}$.
- Independent events. $P(AB)=P(A)P(B)$.
- Complementary events. $P(A')=1-P(A)$.
Example – committee selection
From 8 boys and 5 girls, how many ways to choose a committee of 4 with exactly 2 boys?
- Choose boys: $\binom{8}{2}=28$.
- Choose girls: $\binom{5}{2}=10$.
- Total $=28\times10=280$ ways.
12. Series – arithmetic & geometric
- Arithmetic series. $a, a+d, a+2d,\dots$; $n$‑th term $a_{n}=a+(n-1)d$; sum $S_{n}= \dfrac{n}{2}\bigl(2a+(n-1)d\bigr)=\dfrac{n}{2}(a_{1}+a_{n})$.
- Geometric series. $a, ar, ar^{2},\dots$; $n$‑th term $a_{n}=ar^{\,n-1}$; sum $S_{n}=a\,\dfrac{1-r^{n}}{1-r}$ ( $req1$ ). For $|r|<1$, infinite sum $S_{\infty}= \dfrac{a}{1-r}$.
Example – sum of first 10 terms
Find $S_{10}$ for $2,5,8,\dots$ (common difference $d=3$).
- $a=2$, $a_{10}=2+9\cdot3=29$.
- $S_{10}= \dfrac{10}{2}(2+29)=5\times31=155$.
13. Vectors (plane)
- Notation. $\vec{u}= \langle u_{1},u_{2}\rangle$ (components) or $u_{1}\mathbf{i}+u_{2}\mathbf{j}$.
- Magnitude. $|\vec{u}|=\sqrt{u_{1}^{2}+u_{2}^{2}}$.
- Addition. $\vec{u}+\vec{v}= \langle u_{1}+v_{1},\,u_{2}+v_{2}\rangle$.
- Scalar multiplication. $k\vec{u}= \langle ku_{1},ku_{2}\rangle$.
- Dot product. $\vec{u}\cdot\vec{v}=u_{1}v_{1}+u_{2}v_{2}=|\vec{u}||\vec{v}|\cos\theta$.
- Angle between vectors. $\cos\theta=\dfrac{\vec{u}\cdot\vec{v}}{|\vec{u}||\vec{v}|}$.
- Applications. Find the resultant of forces, determine whether two lines are parallel/perpendicular, work problems.
Worked example – angle between $\vec{a}=\langle3,4\rangle$ and $\vec{b}=\langle5,0\rangle$
- $\vec{a}\cdot\vec{b}=3\cdot5+4\cdot0=15$.
- $|\vec{a}|=5$, $|\vec{b}|=5$.
- $\cos\theta=\dfrac{15}{5\cdot5}=0.6\;\Rightarrow\;\theta=\cos^{-1}0.6\approx53.1^{\circ}$.
14. Basic Calculus (rates & areas)
- Derivative (gradient of a curve). For $y=f(x)$, $f'(x)=\displaystyle\lim_{\Delta x\to0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$.
- Power rule: $\dfrac{d}{dx}x^{n}=nx^{n-1}$.
- Exponential: $\dfrac{d}{dx}a^{x}=a^{x}\ln a$.
- Logarithmic: $\dfrac{d}{dx}\ln x=\dfrac1x$.
- Trigonometric: $\dfrac{d}{dx}\sin x=\cos x$, $\dfrac{d}{dx}\cos x=-\sin x$.
- Integration (area under a curve). $\displaystyle\int f(x)\,dx$ is the antiderivative. For a definite integral $\int_{a}^{b}f(x)\,dx$, the result is the signed area between the curve and the $x$‑axis.
- Common antiderivatives.
- $\int x^{n}\,dx=\dfrac{x^{n+1}}{n+1}+C\;(neq-1)$
- $\int a^{x}\,dx=\dfrac{a^{x}}{\ln a}+C$
- $\int \frac{1}{x}\,dx=\ln|x|+C$
- $\int \sin x\,dx=-\cos x+C$, $\int \cos x\,dx=\sin x+C$
Example – rate of change
If $s(t)=4t^{2}+3t$ (metres), find the speed at $t=5\,$s.
- Velocity $v(t)=\dfrac{ds}{dt}=8t+3$.
- At $t=5$, $v=8\cdot5+3=43\,$m s$^{-1}$.
Example – area under $y=x^{2}$ from $x=0$ to $x=3$
$\displaystyle\int_{0}^{3}x^{2}\,dx=\Bigl[\frac{x^{3}}{3}\Bigr]_{0}^{3}= \frac{27}{3}=9\;$ square units.
15. Summary of Key Formulae
| Topic | Key formulae |
|---|
| Domain / Range | Identify restrictions from denominators, even roots, logs, trig; solve $y=f(x)$ for $x$ to get range. |
| Composite | $(f\circ g)(x)=f(g(x))$, $\operatorname{Dom}(f\circ g)=\{x\in\operatorname{Dom}(g)\mid g(x)\in\operatorname{Dom}(f)\}$. |
| Inverse | Swap $x$ and $y$ in $y=f(x)$, solve for $y$; domain of $f^{-1}$ = range of $f$. |
| Quadratic | $\Delta=b^{2}-4ac$, vertex $(-b/2a,\;f(-b/2a))$, range $[k,\infty)$ or $(-\infty,k]$. |
| Polynomials | Remainder theorem $f(a)=\text{remainder}$; factor theorem $(x-a)$ factor iff $f(a)=0$. |
| Log/Exp | $\log_{a}b=\dfrac{\ln b}{\ln a}$; $a^{\log_{a}x}=x$; $e^{\ln x}=x$. |
| Line | $y=mx+c$, $m_{1}m_{2}=-1$ for perpendicular. |
| Circle | $(x-h)^{2}+(y-k)^{2}=r^{2}$. |
| Circular measure | $s=r\theta$, $A=\frac12 r^{2}\theta$ (θ rad). |
| Trig | SOH‑CAH‑TOA, $\sin^{2}+\cos^{2}=1$, $1+\tan^{2}=\sec^{2}$. |
| Perm/Comb | $P(n,r)=\dfrac{n!}{(n-r)!}$, $C(n,r)=\dfrac{n!}{r!(n-r)!}$. |
| Series | Arithmetic $S_{n}=n/2\,(2a+(n-1)d)$; Geometric $S_{n}=a(1-r^{n})/(1-r)$. |
| Vectors | $\vec{u}\cdot\vec{v}=|\vec{u}||\vec{v}|\cos\theta$. |
| Calculus | Power rule $d/dx\;x^{n}=nx^{n-1}$; $\int a^{x}dx=a^{x}/\ln a$. |
16. Practice Questions (selected)
- Domain & range. Find the domain and range of $h(x)=\dfrac{1}{\sqrt{4-x}}$.
Answer: Domain $x<4$; range $(0,\infty)$.
- Composite domain. $p(x)=\ln(x-1)$, $q(x)=e^{2x}+1$. Determine $\operatorname{Dom}(p\circ q)$.
Answer: $q(x)>1$ for all $x$, so $\operatorname{Dom}= \mathbb R$.
- Inverse verification. Show that $f(x)=\dfrac{x}{x+2}$ is its own inverse. State domain & range.
Answer: $f(f(x))=x$, domain $xeq-2$, range $\mathbb R\setminus\{1