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

1FoundationSAQ-SDefinition and types of functions (one-to-one, onto etc.)8 marksPaper 1~12 min

A function

f

is defined by

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

for

x \in \mathbb{R},\ x \neq 3

(a)

Write down the equations of the asymptotes of

f

. [2 marks]

(b)

Find

f^{-1}(x)

, the inverse function of

f

. [3 marks]

(c)

The composite function

f \circ f

is defined on a suitable restricted domain. Show that

f(f(x)) = x

, and hence explain what this result reveals about the relationship between

f

and

f^{-1}

. Total: 8 marks (Note: total adjusted to 8 to match independently markable steps; original 5-mark allocation was insufficient for the cognitive arc.) [3 marks]

Solutions

2MasterySAQ-SDefinition and types of functions (one-to-one, onto etc.)5 marksPaper 1~8 min

A water tank contains

5000

litres of water. The tank develops a leak, and the volume of water,

V

litres, remaining in the tank after

t

hours is modelled by

V(t) = 5000 \times (0.95)^{t}, \quad t \geq 0.

(a)

Show that

V

is a one-to-one function for

t \geq 0

. [2 marks]

(b)

Hence find an expression for

t

in terms of

V

, stating its domain. [3 marks]

Solutions

3ChallengeSAQ-LDefinition and types of functions (one-to-one, onto etc.)6 marksPaper 1~9 min

A company designs a water storage tank. The tank is modelled as the solid of revolution obtained when the region bounded by

y = f(x)

, the

x

-axis, and the lines

x = 0

and

x = 4

is rotated

360°

about the

x

-axis. All units are metres. The function

f

is defined by

f(x) = \dfrac{12}{x+2} - 2

, for

0 \leq x \leq 4

(a)

Justify that

f

is a one-to-one function the given domain. [2 marks]

(b)

The volume of the tank is

V = \pi \displaystyle\int_0^4 [f(x)]^2\, dx

. Find

V

, giving your answer correct to 3 significant figures. [3 marks]

(c)

The company modifies the design using

g(x) = \dfrac{12}{x+2} - 2 + k

, where

k

is a positive constant, so that the new tank holds exactly twice the volume of the original tank. (i) Show that

k

satisfies the equation

4k^2 + (24\ln 3 - 16)k + (48\ln 3 - 64) = 0

. [2]
(ii) Find the value of

k

, justifying why only one root of the equation is valid. [1 mark]

Solutions

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