learn · audeac — Interactive IB Mathematics

Coverage

Mixed practice over function definition, domain/range, all families (rational/radical/modulus/piecewise), partial fractions, classification (one-to-one, even/odd), composition, inverses, transformations.

Format

18 questions across 4 sections (A: 5, B: 5, C: 4, D: 4 longer). Designed for ~60 min in one sitting; pause and resume any time.

Function test

Vertical line test (graph) ⇔ no two ordered pairs share an

x

.

Natural domain rules

Combine: denominator

\ne 0

, radicand

\ge 0

(even root), argument

>0

(log).

Range from graph

Read

y

-values reached. For shifted-vertex parabolas, range is bounded by the

y

of the vertex.

Rational $\dfrac{ax+b}{cx+d}$

VA:

x=-d/c

. HA:

y=a/c

.

Radical

y = c + k\sqrt{ax+b}

Domain:

ax+b\ge 0

. Range starts at

y=c

; goes up if

k>0

, down if

k<0

.

Modulus equation

|A| = b\;\Rightarrow\;A=\pm b

Modulus inequalities

|A|\le b\;\Leftrightarrow\;-b\le A\le b\;\;\text{ and }\;\;|A|\ge b\;\Leftrightarrow\;A\le -b\;\text{ or }\;A\ge b

Partial fractions — cover-up

For distinct linear factors: cover

(x-r_i)

and substitute

x=r_i

to read off

A_i

.

Even / odd tests

f(-x)=f(x)\;[\text{even}]\;\;\;f(-x)=-f(x)\;[\text{odd}]

One-to-one ⇒ inverse exists

If many-to-one, restrict the domain (e.g. one side of the parabola vertex) before inverting.

Composition

(f\circ g)(x) = f(g(x))

Find $f^{-1}$

Swap

x

and

y

, solve for

y

. Geometric: reflect graph in

y=x

. Domain/range swap.

Self-inverse

f^{-1}(x) = f(x)

Examples:

1/x

,

a-x

,

(x-k)/(x-1)

for certain

k

.

Combined transform

y = a\,f(b(x-h)) + k

a

vert stretch ·

b

horiz ·

h

right ·

k

up.

Reflections

-f(x):\text{ x-axis};\;\;f(-x):\text{ y-axis}

Reciprocal $1/f$

Zeros of

f

⇒ VAs of

1/f

. Min of

f

⇒ max of

1/f

.

$|f(x)|$ vs $f(|x|)$

|f(x)|

: flip below-axis up.

f(|x|)

: replace

x<0

part with mirror of

x>0

part.

Trap — sign inside bracket

f(x-h)

shifts RIGHT by

h

(subtracting

h

moves right). Always counter-intuitive.

Trap — extraneous solutions

Modulus equations: split into cases and CHECK in the original. Squaring or expanding can add false roots.

Trap — domain of composition

D(f\circ g)

=

\{x \in D(g) : g(x) \in D(f)\}

. Don't just take

D(g)

.