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Functions and Their Properties · Paper 3 — Free Maths AI HL Practice Questions

1ChallengeLAQDefinition and types of functions (one-to-one, onto etc.)10 marksPaper 3~15 min

A function

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

is defined by

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

for all

x \in \mathbb{R}

(a)

Prove that

f

is neither injective (one-to-one) nor surjective (onto) over

\mathbb{R}

. [5 marks]

(b)

The function

g

is defined by restricting the domain of

f

[k, \infty)

for some constant

k \in \mathbb{R}

, with the codomain simultaneously restricted to the range of

g

, so that

g : [k, \infty) \longrightarrow \text{range of } g, \qquad g(x) = f(x).

Determine the complete set of values of

k

for which

g

is a bijection (both injective and surjective). Justify your answer fully. [5 marks]

Solutions

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