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Functions and Their Properties · Paper 1 — Free Maths AA HL Practice Questions

1FoundationSAQ-SDefinition and types of functions (one-to-one, onto etc.)s5 marksPaper 1~8 min

The function

f: \mathbb{R} \to \mathbb{R}

is defined by

f(x) = \begin{cases} 2x + 1 & x \geq 0 \\ x^2 + 1 & x < 0 \end{cases}

(a)

Show that

f

is injective (one-to-one). - [3 marks]

(b)

Determine whether

f

is surjective (onto). Justify your answer. [2 marks]

Solutions

2MasterySAQ-SDefinition and types of functions (one-to-one, onto etc.)s5 marksPaper 1~8 min

A function

h: \mathbb{R} \to \mathbb{R}

is defined by

h(x) = \begin{cases} x+1, & x \leq 2 \\ 5-x, & x > 2 \end{cases}

(a)

Show that

h

is not a one-to-one function. [2 marks]

(b)

Show that

h(x) \leq 3

for all

x \in \mathbb{R}

, and hence show that

h

is not an onto function. [2 marks]

(c)

Deduce the range of

h

, justifying that every value in your stated range is achieved. [1 mark]

Solutions

3ChallengeSAQ-LDefinition and types of functions (one-to-one, onto etc.)s8 marksPaper 1~12 min

A function

f: \mathbb{R} \to \mathbb{R}

is defined by

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

(a)

Determine whether

f

is injective (one-to-one). Justify your answer. [3 marks]

(b)

Determine whether

f

is surjective (onto). Justify your answer. [3 marks]

(c)

State a restricted domain

D \subseteq \mathbb{R}

and a restricted codomain

C \subseteq \mathbb{R}

such that

f: D \to C

is a bijection. Hence write down an expression for

f^{-1}(x)

, stating its domain. [2 marks]

Solutions

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