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Functions

The document contains various mathematical problems involving functions, their inverses, and ranges. It includes tasks such as finding the range of a function, explaining the existence of inverses, and solving equations involving composite functions. Additionally, it addresses the properties of specific functions and their inverses, along with the domains and ranges associated with them.

Uploaded by

Tumi Ogunkoya
Copyright
© © All Rights Reserved
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25 views7 pages

Functions

The document contains various mathematical problems involving functions, their inverses, and ranges. It includes tasks such as finding the range of a function, explaining the existence of inverses, and solving equations involving composite functions. Additionally, it addresses the properties of specific functions and their inverses, along with the domains and ranges associated with them.

Uploaded by

Tumi Ogunkoya
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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7

5 f : x 7 (2x + 3) 2 for x 2 0

(a) Find the range of f. [1]

(b) Explain why f has an inverse. [1]

(c) Find f -1. [3]

(d) State the domain of f -1. [1]

(e) Given that g : x 7 ln (x + 4) for x 2 0, find the exact solution of fg (x) = 49 . [3]

© UCLES 2020 0606/12/M/J/20 [Turn over


10

1
10 (a) g (x) = 3 + for x H 1.
x
(i) Find an expression for g -1 (x) . [2]

(ii) Write down the range of g -1 . [1]

(iii) Find the domain of g -1 . [2]

© UCLES 2020 0606/22/F/M/20


10

10 (a) The function f is defined by f (x) = 1 + x 2 , for all real values of x. The graph of y = f(x) is
given below.

y = 1 + x2

O x

(i) Explain, with reference to the graph, why f does not have an inverse. [1]

(ii) Find f 2 (x) . [2]

(b) The function g is defined, for x > k, by g (x) = 1 + x 2 and g has an inverse.

(i) Write down a possible value for k. [1]

(ii) Find g -1 (x) . [2]

© UCLES 2018 0606/22/F/M/18


12

11 The functions f and g are defined by


x2 - 2
f (x) = for x H 2 ,
x

x2 - 1
g (x) = for x H 0 .
2

(i) State the range of g. [1]

(ii) Explain why fg(1) does not exist. [2]

c
(iii) Show that gf (x) = ax 2 + b + , where a, b and c are constants to be found. [3]
x2

© UCLES 2017 0606/22/F/M/17


13

(iv) State the domain of gf. [1]

x+ x2 + 8
(v) Show that f -1 (x) = . [4]
2

© UCLES 2017 0606/22/F/M/17 [Turn over


7

6 The function f is defined by f (x) = 2 - x + 5 for - 5 G x 1 0 .

(i) Write down the range of f. [2]

(ii) Find f -1 (x) and state its domain and range. [4]

4
The function g is defined by g (x) = for - 5 G x 1 - 1 .
x
(iii) Solve fg (x) = 0 . [3]

© UCLES 2016 0606/13/M/J/16 [Turn over


10

9 (a) It is given that g (x) = 6x 4 + 5 for all real x.

(i) Explain why g is a function but does not have an inverse. [2]

(ii) Find g 2 (x) and state its domain. [2]

It is given that h (x) = 6x 4 + 5 for x G k .

(iii) State the greatest value of k such that h-1 exists. [1]

(iv) For this value of k, find h-1(x). [3]

© UCLES 2019 0606/22/F/M/19

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