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

1FoundationSAQ-SDefinition and types of functions (one-to-one, onto etc.)6 marksPaper 1~9 min

A function

f

is defined by

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

, for

x \neq 2

(a)

Write down the range of

f

. [1 mark]

(b)

Find

f^{-1}(x)

, stating its domain. [3 marks]

(c)

Hence determine the value of

x

for which

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

, and explain why there is only one such value. [2 marks]

Solutions

2FoundationSAQ-SDefinition and types of functions (one-to-one, onto etc.)4 marksPaper 1~6 min

The graph of

f

is shown below. The domain of

f

-4 \leq x \leq 4

. -

(a)

(i) State whether

f

is a one-to-one function. [1] -
(ii) Justify your answer to (a)(i). - [1 mark]

(b)

Find the range of

f

. [3 marks]

Solutions

3MasterySAQ-SDefinition and types of functions (one-to-one, onto etc.)5 marksPaper 1~8 min

Consider the function

g(x) = x^2 - 4x + 5

, defined for

x \in \mathbb{R}

(a)

Show that

g

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

(b)

The domain of

g

is restricted to

x \geq k

so that

g

becomes one-to-one. Write down the smallest possible value of

k

. [1 mark]

(c)

Hence find the range of

g

for the restricted domain

x \geq k

. [2 marks]

Solutions

4ChallengeSAQ-LDefinition and types of functions (one-to-one, onto etc.)6 marksPaper 1~9 min

A function

f

is defined by

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

for

x \in \mathbb{R},\ x \neq 1

(a)

Determine whether

f

is a one-to-one function. [3 marks]

(b)

Write down the range of

f

. [1 mark]

(c)

A second function

g

is defined by

g(x) = \dfrac{ax+b}{cx+d}

, where

a,b,c,d \in \mathbb{R}

and

c \neq 0

. The graph of

g

has a vertical asymptote at

x = 2

and a horizontal asymptote at

y = 3

. The point

(0,\ 1)

lies on the graph of

g

. (i) Find the values of

a

b

c

, and

d

. [2]
(ii) The composite function

h = g \circ f

is defined on the domain of

f

. Find

h(x)

, simplifying your answer fully, and state its range. [2 marks]

Solutions

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