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Chapter 8 Functions

14 The function f is defined as f : x  (^) x^2 −^ x 6 x ≠ 6 (a) Find f(10)

...................................................... (1) (b) Express the inverse function f −1^ in the form f −1^ : x  …

f −1^ : x  ...................................................... (3) (Total for Question 14 is 4 marks)

1. June 2022 1H # 14

15 The functions f and g are such that (^) f

g

x x x (^) xx

(a) State the value of x that cannot be included in any domain of g ....................................................... (1) (b) Find gf( x ) Simplify your answer.

gf( x ) = ....................................................... (2) (c) Express the inverse function g−1^ in the form g−1( x ) = ...

g−1( x ) = ....................................................... (3) (Total for Question 15 is 6 marks)

3 ) November 2021 2H # 15

19 f( x ) = x^2 – 4 g( x ) = 2 x + 1 Solve fg( x ) > 0 Show clear algebraic working.

....................................................... (Total for Question 19 is 4 marks)

4 ) June 2022 2HR # 19

23 The functions f and g are such that f( x ) = x + 25 g( x ) = x^2 – 12 x The function h is such that h( x ) = fg( x ) The domain of h is { x : x  6} Express the inverse function h–1^ in the form h–1^ ( x ) = ...

h –1( x ) = ....................................................... (Total for Question 23 is 4 marks)

6 ) January 2022 2HR # 23

24 The functions f and g are defined as f( x ) = 5 x^2 − 10 x + 7 where x  1 g( x ) = 7 x − 6 (a) Find fg(2)

....................................................... (2) (b) Express the inverse function f −1^ in the form f −1^ ( x ) = ...

f −1^ ( x ) = ....................................................... (4) (Total for Question 24 is 6 marks)

7 ) June 2021 1H # 24

17 The functions f and g are defined as f( x ) = x^2 + 6 g( x ) = x – 10 (a) Find fg(3) ...................................................... (2) (b) Solve the equation fg( x ) = f( x ) Show clear algebraic working.

...................................................... (3) The function h is defined as h( x ) = 2 xx^ −^4 (c) State the value of x that cannot be included in the domain of h

...................................................... (1) (d) Express the inverse function h –1^ in the form h –1^ ( x ) = ...

h–1( x ) = ...................................................... (3) (Total for Question 17 is 9 marks)

9 ) January 2021 2HR #

18 The function f is such that f( x ) = kx where x ≠ 0 and k is an integer.

(a) Express the inverse function f −1^ in the form f −1( x ) = ...

f −1( x ) = ...................................................... (1) The function g is such that g( x ) = 2 – 3 x^4 where x ≠ 0 The function h is such that h( x ) = (^23) −^ x x where x ≠ 2 (b) (i) Find g(−2)

...................................................... (1) (ii) Express the composite function hg in the form hg( x ) = ... Give your answer in its simplest form.

hg( x ) = ...................................................... (2) (Total for Question 18 is 4 marks)

10 ) January 2023 1H

16 The function f is such that f( x ) = (^3) x^2 − 5 where x ≠ (^53) (a) Find f  13 

...................................................... (1) (b) Find f –1( x )

f –1( x ) = ...................................................... (2) The function g is such that g( x ) = 5 x^2 – 20 x + 23 (c) Express g( x ) in the form a ( x – b ) 2 + c

...................................................... (3) (Total for Question 16 is 6 marks)

12 ) January 2023 1HR

17 The functions g and h are such that g( x ) = (^2) x^11 − 5 h( x ) = x^2 + 4 x  0 (a) What value of x must be excluded from any domain of g?

....................................................... (1) (b) Solve gh( x ) = 1

....................................................... (3) (Total for Question 17 is 4 marks)

13 ) June 2023 1H

17 f( x )  (^2) xx  4 g( ) x  3 x  1

Given that fg( k ) = 2 work out the value of k

k = .............................................................. (Total for Question 17 is 3 marks)

15 ) June 2024 2HR # 17

25 The function f is such that f  x   2 x^2  24 x  7 where x  6

Find the inverse function f ^1  x 

f ^1  x  .............................................................................................

(Total for Question 25 is 4 marks)

16 ) November 2024 1H # 25