1.1 Functions: Language & Notation

Quick Notes & Formulae

Essential concepts for solving function problems.

Function Definition

A mapping where every input element

x

maps to exactly one output element

y

Domain vs Range

D o m a i n

: Set of all possible inputs (x-values).

R a n g e

: Set of all resulting outputs (y-values), also called the image set.

One-One Function

Matches each output to exactly one input. passes the <strong>Horizontal Line Test</strong>. Required for an inverse function to exist.

Many-One Function

Multiple inputs can map to the same output (e.g.,

y = x^{2}

). Not one-one, so no inverse unless domain is restricted.

Completed Square

a (x - h)^{2} + k

. Vertex at

(h, k)

. Useful for finding range of quadratics.

Composite Function

f g (x) = f (g (x))

. Apply

g

first, then

f

. Domain of

f

must accept range of

g

Past Paper Style Questions

Question 1: Function Mapping, Domain & Range

A mapping is defined from set

A = {- 2, - 1, 0, 1, 2, 3}

to set

B = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

, with

f : x \mapsto x^{2} + 1

(a) Write down

f (- 2), f (0), f (3)

. [2]

(b) List the range (image set) of

f

for the given domain

A

. [2]

f

is one–one or many–one on domain

A

. Give a reason. [1]

(d) Find a restricted domain

A^{'} \subseteq A

such that

f

becomes one–one. [1]

Question 2: Graph to Domain/Range & One-one Test

The graph of

y = f (x)

consists of:

a straight line segment from $(- 2, 3)$ to $(1, 0)$ , and
a curve $y = (x - 1)^{2}$ for $1 \leq x \leq 4$ .

(a) State the domain of

f

. [1]

(b) State the range of

f

. [2]

f

is one–one on its domain. [1]

(d) State a domain restriction that makes

f

one–one, and write the new range. [2]

Question 3: Function Notation and Image Set

A function is defined by

f (x) = \frac{2 x - 3}{x + 1}

(a) Evaluate

f (0), f (2), f (- 3)

. [3]

(b) State the domain of

f

for real

x

. [1]

f (x) \neq 2

for all real

x

in the domain. [2]

(d) Hence state the range of

f

. [1]

Question 4: Many-one vs One-one using Algebra

Let

f (x) = x^{2} - 4 x + 1

for

x \in R

(a) Write

f (x)

in completed square form. [2]

(b) Find the minimum value of

f (x)

. [1]

f

. [1]

(d) Explain why

f

is many–one. [1]

(e) State a restricted domain that makes

f

one–one. [1]

Question 5: Mapping Diagram Interpretation

A mapping diagram

(domain \to codomain)

shows:

$1 \mapsto 4$
$2 \mapsto 4$
$3 \mapsto 5$
$4 \mapsto 7$
$5 \mapsto 7$

(a) Is this relation a function? Give a reason. [1]

(b) State whether it is one–one or many–one. [1]

(d) One value in the domain is removed so the mapping becomes one–one. State one possible value removed and justify briefly. [2]

Question 6: Piecewise Function

A function

f

is defined by:

f (x) = {\begin{cases} 3 - x, & - 1 \leq x < 2 \\ \sqrt{x - 2}, & 2 \leq x \leq 6 \end{cases}

(a) State the domain of

f

. [1]

(b) Find the range of

f

. [2]

f

is one–one on its domain. [1]

(d) Find a domain restriction so that

f

becomes one–one. [1]

Question 7: Logarithmic Function

The function

f

is given by

f (x) = \ln (x - 1) + 2

(a) State the domain of

f

. [1]

(b) State the range of

f

. [1]

y = f (x)

, showing asymptotes. [2]

(d) Explain whether

f

is one–one. [1]

Question 8: Function Test

Consider the curve

x = y^{2} - 2 y

(a) Explain why this relation does not define y as a function of x over reals. [1]

(b) Rearrange to express y in terms of x. [2]

Question 9: Modulus Function

Let

f (x) = | 2 x - 5 |

for

- 1 \leq x \leq 6

(a) Find

f (- 1), f (2.5), f (6)

. [2]

(b) State the domain and range. [2]

f

one–one on

[- 1, 6]

? Explain. [1]

(d) Give a restricted domain interval where

f

is one–one. [1]

Question 10: Transformed Graph

Basic graph

y = \sqrt{x}

is transformed to

f (x) = 3 - 2 \sqrt{x + 1}

(a) State the domain of

f

. [1]

(b) State the range of

f

. [2]

x = - 1

. [1]

(d) Is

f

one–one? [1]