FUNCTIONS

–6– M15/5/MATHL/HP1/ENG/TZ1/XX
1.
5. [Maximum mark: 6]

The functions f and g f (x) ax2 bx c , x and

g (x) p sin x qx r , x where a , b , c , p , q , r are real constants.

(a) Given that f is an even function, show that b 0 . [2]

(b) Given that g r. [2]

The function h is both odd and even, with domain .

(c) Find h (x) . [2]

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–7– M15/5/MATHL/HP1/ENG/TZ1/XX
2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6. [Maximum
. . . . .mark:
. . . . . 7]
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2.
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. .,.x. . . . ., .x. . .1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . .f . . . . . . . . . . . . .f. (.x.).=. .3.x.−. 2
A function
. . . . . . . . . . . . . . . . . . . . . . . .2.x. −. 1. . . . . . . . . . . .2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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(a) Find an expression for f –1 (x) . [4]
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B
(b) . . . . . .that
Given . . . .f.(x)
. . .can
. . . .be
. . written
. . . . . . in
. .the . . . . .f. (.x.).=. .A. +. . . . . . . . . . . . . . . . . . . . . . . . . .
. . . form
2x −1
constants A and B . [2]

3x − 2
(c) Hence, write down ∫ 2 x − 1 dx . [1]
3.

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16EP06

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16EP07
 – 10 – M15/5/MATHL/HP1/ENG/TZ1/XX
4.
9. [Maximum mark: 9]

π
The functions f and g f ( x) = 2 x + ,x and g (x) 3sin x 4, x .
5

 π
(a) Show that g f (x) = 3 sin  2 x + +4. [1]
 5

(b) Find the range of g f . [2]

 3π   3π 
(c) Given that g f  =7 x , greatergthan
f  , for
= 7which
 20   20 
g f (x) 7. [2]

(d) The graph of y g f (x) can be obtained by applying four transformations to the graph
of y sin x . State what the four transformations represent geometrically and give the
order in which they are applied. [4]

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(This question continues on the following page)

 5. –8– M16/5/MATHL/HP1/ENG/TZ1/XX

7. [Maximum mark: 8]

7
Sketch on the same axes the curve y
(a)
x 4 and the line y x 2 , clearly
indicating any axes intercepts and any asymptotes. [3]

7
(b) Find the exact solutions to the equation x 2 x 4 . [5]

–3– M16/5/MATHL/HP1/ENG/TZ2/XX
6.
2. [Maximum mark: 5]

3x 2
The function f f (x) ,x ,x 1.
x 1

Sketch. .the
. . .graph
. . . . .of
. . y. . . f. .(x)
. .,. clearly
. . . . . .indicating
. . . . . . . . .and
. . . stating
. . . . . . the
. . . equations
. . . . . . . . . of
. . any
. . . .asymptotes
............
and the
. . coordinates
. . . . . . . . . . of
. . any
. . . .axes
. . . . intercepts.
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. . . . . ....... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ....... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ....... .. .. .. .. .. .

. . . . . ....... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ....... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ....... .. .. .. .. .. .
. . . . . ....... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ....... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ....... .. .. .. .. .. .
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 7.
–7– N17/5/MATHL/HP1/ENG/TZ0/XX

6. [Maximum mark: 9]

1 − 3x
(a) Sketch the graph of y = , showing clearly any asymptotes and stating the
x−2
coordinates of any points of intersection with the axes. [4]
y
10

x
–5 0 5 10

–5

–10

1 − 3x
(b) Hence or otherwise, solve the inequality < 2. [5]
x−2

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 –7– M17/5/MATHL/HP1/ENG/TZ1/XX
8.
6. [Maximum mark: 5]

Consider the graphs of y | x | and y |x| b , where b .

(a) Sketch the graphs on the same set of axes. [2]

b. [3]

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–3– M17/5/MATHL/HP1/ENG/TZ2/XX
9.
..........................................................................
2. [Maximum mark: 6]

The function f f (x) 2x3 5 , 2 x 2.

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

(b) Find an expression for f 1(x) . [2]

1
(c) Write down the domain and range of f . [2]

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. . . . over
 –3– M18/5/MATHL/HP1/ENG/TZ2/XX
10.
2. [Maximum mark: 7]

x
(a) Sketch the graphs of y + 1 and y |x 2| on the following axes. [3]
2

y
6

5
4
3

1
x
4 2 0 2 4 6 8
1

x
(b) Solve the equation + 1 = | x − 2|. [4]
2

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 11.
–6– M19/5/MATHL/HP1/ENG/TZ2/XX

5. [Maximum mark: 8]

x−4
(a) Sketch the graph of y = , stating the equations of any asymptotes and the
2x − 5

x
2 1 0
1

x−4
(b) Consider the function f : x → .
2x − 5
Write down

(i) the largest possible domain of f ;

(ii) the corresponding range of f . [3]

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Do not write solutions on this page.

Section B

Answer all questions in the answer booklet provided. Please start each question on a new page.
12.
10. [Maximum mark: 17]

3x
The function f f ( x) = , x∈ , x ≠ 2.
x−2

(a) Sketch the graph of y f (x) , indicating clearly any asymptotes and points of
intersection with the x and y axes. [4]

1
(b) Find an expression for f (x) . [4]

(c) Find all values of x for which f (x) f 1 (x) . [3]

3
(d) Solve the inequality f ( x) . [4]
2

(e) Solve the inequality f ( x ) < 32 . [2]

11. [Maximum mark: 16]

π x +1
Consider the functions f ( x) = tan x , 0 ≤ x < and g ( x) = , x ∈ , x ≠ 1.
2 x −1

(a) Find an expression for g f (x) , stating its domain. [2]

sin x + cos x
(b) Hence show that g f ( x) = . [2]
sin x − cos x

dy
(c) Let y g f (x) at the point on the graph of y g f (x)
dx
π
where x = , expressing your answer in the form a + b 3 , a , b ∈ . [6]
6

(d) Show that the area bounded by the graph of y g f (x) , the x-axis and the lines
π
x
6
(
0 and x = is ln 1 + 3 . ) [6]

Turn over

12EP11
Do not(e)
writeSolve
13.
the inequality
solutions f( x
on this page. ) < 32 . [2]
11. [Maximum mark: 16] – 12 – M17/5/MATHL/HP1/ENG/TZ1/XX
11. [Maximum mark: 17]
Do not write solutions on this page. π x +1
13. themark:
11. Consider
[Maximum 16] 2 f ( x) = tan x , 0 ≤ x <
functions and2 g ( x) = , x∈ , x ≠ 1.
(a) (i) Express x 3x 2 in the form (x 2 h) k . x −1
11. [Maximum
13. mark: 17]
π x +1
Consider
(a) (ii) the
Find functionsx2 f for
Factorize
an expression x) g= tan
(3x x , 0, stating
2 .f (x) ≤ x < its
and g ( x) =
domain. , x ∈ , x ≠ 1. [2]
[2]
Express x2 3x 2 in the form (x 2 h) k . x − 1
2
(a) (i)

2 1 x + cos xits domain.

sin
(ii) the
(a) Hence
Find
Consider
(b) thatxfg( xfor
Factorize
an expression
function
show )3x
f (gx)22=.f (x) , stating
,x . ,x 2, x 1. [2]
[2]
[2]
x 3 xx − 2cos x
sin
1 x + cos x
sin
Consider
(b)
(b) thethe
Hence
Sketch function
showgraph g( x)ff(x)
that fof ( x), 2=indicating ,onx it.dythe
, x equations
2, x of 1the
. asymptotes, [2]
(c) Let y g f (x) x sin 3 xx − 2cos x at the point on the graph of y g f (x)
the coordinates of the y-intercept and the dxlocal maximum. [5]
π
(b) where = graph
Sketchx the of f (x) your
, expressing , indicating ytheform
answeroninitdthe a + b of3the
equations , a , asymptotes,
b∈ . [6]
(c) Let y g 6f 1(x) of the1 y-intercept1and the local
the coordinates at the point on the graph of y g f (x)
maximum. [5]
(c) Show that π − = 2 .dx [1]
x + 1 x + 2 x + 3 x + 2
where x = , expressing your answer in the form a + b 3 , a , b ∈ . [6]
(d) Show that the 6 1area bounded
π 1 1 graph of y g f (x) , the x-axis and the lines
by the
(c) xShow
0 and x+
−
(
thatx = is ln 1 + 3 1.2 =
) .
6 1 x + p2 if x0 f+(3xx) d+x 2 ln( p) .
[1]
[6]
[4]
(d) Show that the area bounded by the graph of y g f (x) , the x-axis and the lines
π
x 0 and x = is ln 1 + 3 1. ( )
6 of y pf (if| x |0) .f ( x) dx ln( p) .
[6]
[4][2]
(d) Sketch the graph
(e)

(f)(e) Determine
Sketch thethe area
graph (| x |) . enclosed between the graph of y
ofofy thef region f (| x |) , the x-axis [2]
and the lines with equations x 1 and x 1 . [3]
(f) Determine the area of the region enclosed between the graph of y f (| x |) , the x-axis
Turn over
and the lines with equations x 1 and x 1 . [3]
12EP11
Turn over

12EP11

16EP12

16EP12
Do not write solutions on this page.

Section B
14.
Answer all questions in the answer booklet provided. Please start each question on a new page.
14.
9. [Maximum mark: 17]

Consider the function f f (x) x2 a2 , x where a is a positive constant.

(a) Showing any x and y intercepts, any maximum or minimum points and any
asymptotes, sketch the following curves on separate axes.

(i) y f (x) ;

1
(ii) y ;
f ( x)

1
(iii) y . [8]
f ( x)

(b) Find f (x) cos x dx. [5]

The function g g ( x) x f ( x) for | x | a.

g'(x) explain why g is an increasing function. [4]

12EP10
Do not write solutions on this page.
The function h h( x ) x , for x 0.
Section B
(c) State the domain and range of h g . [4]
Answer all questions in the answer booklet provided. Please start each question on a new page.
15.
10. [Maximum
11. [Maximum mark:
mark: 20]
12]

Consider the function f x1 ax3 bx2 cx d , where x and a , b , c , d .

(a) Show that log r 2 x log r x where r , x . [2]
(a) (i) Write down an2 expression for f x .

It is given
(ii) that log2 ygiven
Hence, logthat
4 x flog1 4does
2x not
0 . exist, show that b2 3ac 0. [4]

(b) Express y in terms of x . Give 1your answer in the form y pxq , where p , q are
(b) Consider the function g ( x ) x3 3 x 2 6 x 8 , where x .
constants. 2 [5]
(i) Show that g exists.
1

The region R , is bounded by the graph of the function found in part (b), the x-axis, and the
lines x(ii) 1 and
g x x can be written in1 the
where . The area
form x R2is3 2q ,. where p , q
p of .
Find the value of p and the value of q .
(c) Find the value of .1 [5]
(iii) Hence find g x . [8]

The graph of y g x may be obtained by transforming the graph of y x3 using a

sequence of three transformations.

(c) State each of the transformations in the order in which they are applied. [3]

(d) Sketch the graphs of y g x and y g 1 x on the same set of axes, indicating the
points where each graph crosses the coordinate axes. [5]

11. [Maximum mark: 15]

Consider the curve C defined by y2 sin xy , y 0.

12EP11
dy y cos xy
(a) Show that . [5]
dx 2 y x cos xy
dy
(b) Prove that, when 0, y 1. [5]
dx
dy
(c) Hence find the coordinates of all points on C , for 0 x 4 , where 0. [5]
dx

Turn over
16EP13
 – 11 – M18/5/MATHL/HP1/ENG/TZ2/XX

Do not write solutions on this page.

16.
10. [Maximum mark: 14]

ax + b d
The function f f ( x) = , for x ,x
cx + d c

(a) Find the inverse function f , stating its domain. [5]

2x − 3
The function g g ( x) = , x∈ , x ≠ 2 .
x−2

B N20/5/MATHL/HP1/ENG/TZ0/XX
(b) (i) Express g(x) in the form A + where A , B are constants.
x−2

(ii) Sketch the graph of y g(x) . State the equations of any asymptotes and the
coordinates of any intercepts with– the
13 –axes. [5]
8820 – 7201

Do not write solutions on this page.

The function h h( x ) x , for x 0.
Section B
(c) State the domain and range of h g . [4]
Answer all questions in the answer booklet provided. Please start each question on a new page.

10. [Maximum
11. [Maximum mark:
mark: 20]
12]

Consider the function f x1 ax3 bx2 cx d , where x and a , b , c , d .

(a) Show that log r 2 x log r x where r , x . [2]
(a) (i) Write down an2 expression for f x .

It is given
(ii) that log2 ygiven
Hence, logthat
4 x flog1 4does
2x not
0 . exist, show that b2 3ac 0. [4]

(b) Express y in terms of x . Give 1your answer in the form y pxq , where p , q are
(b) Consider the function g ( x ) x3 3 x 2 6 x 8 , where x .
constants. 2 [5]
(i) Show that g 1 exists.
The region R , is bounded by the graph of the function found in part (b), the x-axis, and the
lines x(ii) 1 and
g x x can be written in1 the
where . The area
form x R2is3 2q ,. where p , q
p of .
Find the value of p and the value of q .
(c) Find the value of .1 [5]
(iii) Hence find g x . [8]

The graph of y g x may be obtained by transforming the graph of y x3 using a

sequence of three transformations.

(c) State each of the transformations in the order in which they are applied. [3]

(d) Sketch the graphs of y g x and y g 1 x on the same set of axes, indicating the
points where each graph crosses the coordinate axes. [5]

11. [Maximum mark: 15]

Consider the curve C defined by y2 sin xy , y 0.

12EP11
dy y cos xy
(a) Show that . [5]