FUNCTIONS
–6–
1.
5.
M15/5/MATHL/HP1/ENG/TZ1/XX
[Maximum mark: 6]
The functions f and g
g (x) p sin x qx r , x
f (x) ax2 bx c , x
and
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
[2]
The function h is both odd and even, with domain
(c)
r.
.
Find h (x) .
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[2]
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–7–
M15/5/MATHL/HP1/ENG/TZ1/XX
2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.
. . . . .mark:
. . . . . 7]
...........................................................
[Maximum
2.
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. . . . .f . . . . . . . . . . . . .f. (.x.).=. .3.x.−. 2
. .,.x. . . . ., .x. . .1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A function
. . . . . . . . . . . . . . . . . . . . . . . .2.x. −. 1
. . . . . . . . . . . .2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(a)
.....................................................................
Find an expression for f –1 (x) .
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[4]
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(b)
B
2x −1
. . . . . .that
. . . .f.(x)
. . .can
. . . .be
. . written
. . . . . . in
. .the
. . . form
. . . . .f. (.x.).=. .A. +. . . . . . . . . . . . . . . . . . . . . . . . . .
Given
constants A and B .
(c)
3.
Hence, write down
[2]
3x − 2
∫ 2 x − 1 dx .
[1]
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16EP06
Turn over
16EP07
– 10 –
4.
9.
M15/5/MATHL/HP1/ENG/TZ1/XX
[Maximum mark: 9]
f ( x) = 2 x +
The functions f and g


(a)
Show that g f (x) = 3 sin  2 x +
(b)
Find the range of g f .
(c)
Given that g
g f (x)
(d)
7.
π
,x
5
and g (x)
3sin x
4, x
.
π
+4.
5
 3π 
f  =7
 20 
[1]
[2]
 3π 
f  , for
= 7which
x , greatergthan
 20 
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.
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(This question continues on the following page)
[2]
[4]
7.
M16/5/MATHL/HP1/ENG/TZ1/XX
–8–
5.
[Maximum mark: 8]
(a)
7
x 4 and the line y
Sketch on the same axes the curve y
indicating any axes intercepts and any asymptotes.
(b)
Find the exact solutions to the equation x
2.
[5]
M16/5/MATHL/HP1/ENG/TZ2/XX
–3–
6.
2 , clearly
[3]
7
x 4 .
2
x
[Maximum mark: 5]
The function f
f (x)
3x 2
,x
x 1
,x
1.
. . .graph
. . . . .of
. . y. . . f. .(x)
. .,. clearly
. . . . . .indicating
. . . . . . . . .and
. . . stating
. . . . . . the
. . . equations
. . . . . . . . . of
. . any
. . . .asymptotes
............
Sketch. .the
and the
. . coordinates
. . . . . . . . . . of
. . any
. . . .axes
. . . . intercepts.
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. . . . . ....... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ....... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ....... .. .. .. .. .. .
. . . . . ....... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ....... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ....... .. .. .. .. .. .
. . . . . ....... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ....... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ....... .. .. .. .. .. .
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7.
6.
–7–
N17/5/MATHL/HP1/ENG/TZ0/XX
[Maximum mark: 9]
(a)
Sketch the graph of y =
1 − 3x
, showing clearly any asymptotes and stating the
x−2
[4]
coordinates of any points of intersection with the axes.
y
10
5
x
–5
0
5
10
–5
–10
(b)
Hence or otherwise, solve the inequality
1 − 3x
< 2.
x−2
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[5]
6.
M17/5/MATHL/HP1/ENG/TZ1/XX
–7–
8.
[Maximum mark: 5]
Consider the graphs of y
(a)
| x | and y
|x|
b , where b
.
Sketch the graphs on the same set of axes.
[2]
b.
[3]
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M17/5/MATHL/HP1/ENG/TZ2/XX
–3–
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]
(c)
Write down the domain and range of f
1
.
[2]
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Turn
. . . . over
–3–
10.
2.
M18/5/MATHL/HP1/ENG/TZ2/XX
[Maximum mark: 7]
(a)
Sketch the graphs of y
x
+ 1 and y
2
|x
2| on the following axes.
[3]
y
6
5
4
3
2
1
4
2
0
2
4
6
8
x
1
2
3
4
(b)
Solve the equation
x
+ 1 = | x − 2|.
2
[4]
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Turn over
11.
5.
–6–
M19/5/MATHL/HP1/ENG/TZ2/XX
[Maximum mark: 8]
(a)
Sketch the graph of y =
x−4
, stating the equations of any asymptotes and the
2x − 5
y
5
4
3
2
1
2
1
x
0
1
2
3
4
5
(b)
Consider the function f : x →
x−4
.
2x − 5
Write down
(i)
the largest possible domain of f ;
(ii)
the corresponding range of f .
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[3]
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]
f ( x) =
The function f
(a)
11.
3x
, x∈ , x ≠ 2.
x−2
Sketch the graph of y f (x) , indicating clearly any asymptotes and points of
intersection with the x and y axes.
1
[4]
(b)
Find an expression for f
(x) .
(c)
Find all values of x for which f (x)
(d)
Solve the inequality
f ( x)
3
.
2
[4]
(e)
Solve the inequality f
( x ) < 32 .
[2]
[4]
f 1 (x) .
[3]
[Maximum mark: 16]
Consider the functions f ( x) = tan x , 0 ≤ x <
(a)
Find an expression for g
(b)
Hence show that g
(c)
Let y
g
where x =
(d)
x +1
π
, x ∈ , x ≠ 1.
and g ( x) =
x −1
2
f (x) , stating its domain.
f ( x) =
sin x + cos x
.
sin x − cos x
[2]
dy
at the point on the graph of y
dx
f (x)
g
f (x)
π
, expressing your answer in the form a + b 3 , a , b ∈ .
6
Show that the area bounded by the graph of y
x
[2]
π
0 and x = is ln 1 + 3 .
6
(
)
g
[6]
f (x) , the x-axis and the lines
[6]
Turn over
12EP11
f( x
the inequality
Do not(e)
writeSolve
solutions
on this page.
13.
11. [Maximum
mark: 16]
) < 32 .
[2]
M17/5/MATHL/HP1/ENG/TZ1/XX
– 12 –
11. [Maximum mark: 17]
x +1
Do not write solutions on this page.
π
, x∈
functions
and2 g ( x) =
13. themark:
11. Consider
[Maximum
16] 2 f ( x) = tan x , 0 ≤ x <
(a) (i)
Express x 3x 2 in the form (x
x −1
2 h) k .
11. [Maximum
mark: 17]
13.
π
, x ≠ 1.
x +1
g ( x) =
, x ∈ , x ≠ 1.
Consider
functionsx2 f for
and
(3x
x) g= tan
x , 0, stating
≤ x < its
(ii) the
Factorize
2 .f (x)
(a)
Find
an
expression
domain.
2
(a) (i)
Express x2 3x 2 in the form (x
2 h) k . x − 1
[2]
[2]
2
1 x + cos xits domain.
sin
(ii) the
Factorize
(a) Hence
Find
an
expression
Consider
function
)3x
,x . ,x
2, x
(b)
show
thatxfg( xfor
f (gx)22=.f (x) , stating
[2]
[2]
[2]
1.
3 xx − 2cos x
sin
1 x + cos x
sin
Consider
thethe
function
. asymptotes,
x it.dythe
, x equations
2, x of 1the
(b)
Hence
show
that fof
g( x)ff(x)
( x), 2=indicating ,on
(b)
Sketch
graph
3 xx − 2cos x
x sin
y
g
f
(x)
at
the
point
on
the
graph of y g f (x)
(c) Let
the coordinates of the y-intercept and the
dxlocal maximum.
π
(b) where
Sketchx the
of f (x) your
, indicating
equations
ytheform
= graph
a + b of3the
, a , asymptotes,
b∈ .
, expressing
answeroninitdthe
y g 6f 1(x) of the1 y-intercept1and the local
at the
point on the graph of y g f (x)
(c) Let
the coordinates
maximum.
−
= 2
(c) Show that π
.dx
3
2
x
+
1
x
+
2
x
+
x
+
where x = , expressing your answer in the form a + b 3 , a , b ∈ .
(d) Show that the
by the
6 1area bounded
1
1 graph of y g f (x) , the x-axis and the lines
π
−
=
(c) xShow
thatx = is ln 1 + 3 1.2
.
0 and
x+
6 1 x + p2 if x0 f+(3xx) d+x 2 ln( p) .
(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) .
(d) Sketch the graph
(e)
x
(
)
(
)
(f)(e) Determine
area
Sketch thethe
graph
ofofy thef region
(| x |) . enclosed between the graph of y
and the lines with equations x
1 and x 1 .
(f)
Determine the area of the region enclosed between the graph of y
and the lines with equations x
1 and x 1 .
12EP11
12EP11
16EP12
16EP12
[2]
[5]
[6]
[5]
[1]
[6]
[1]
[6]
[4]
[6]
[4][2]
f (| x |) , the x-axis
[2]
[3]
f (| x |) , the x-axis
Turn over
[3]
Turn over
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.
9.
14.
[Maximum mark: 17]
Consider the function f
(a)
(b)
f (x)
x2 a2 , x
where a is a positive constant.
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) ;
(ii)
y
1
;
f ( x)
(iii)
y
1
.
f ( x)
[8]
Find f (x) cos x dx.
The function g
[5]
g ( x)
x f ( x) for | x |
a.
g'(x) explain why g is an increasing function.
12EP10
[4]
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 .
Answer all questions in the answer booklet provided. Please start each question on a new page.
[4]
15.
10. [Maximum
[Maximum mark:
mark: 20]
11.
12]
Consider the function f x1 ax3 bx2 cx d , where x
log r x where r , x
(a) Show that log r 2 x
.
(a) (i)
Write down an2 expression for f x .
and a , b , c , d
[2]
It is given
log2 ygiven
logthat
2x not
0 . exist, show that b2
(ii) that
Hence,
flog1 4does
4 x
(b)
(b)
.
3ac
0.
[4]
Express y in terms of x . Give 1your
answer in the form y pxq , where p , q are
x3 3 x 2 6 x 8 , where x
Consider the function g ( x )
.
constants.
2
[5]
(i)
Show that g 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
where
. The
area
g x x can be
written in1 the
form
p of
x R2is3 2q ,. where p , q
.
Find the value of p and the value of q .
1
(c)
Find the value of .1
(iii) Hence find g x .
[5]
[8]
The graph of y g x may be obtained by transforming the graph of y
sequence of three transformations.
11.
x3 using a
(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]
[Maximum mark: 15]
Consider the curve C defined by y2
y cos xy
sin xy , y
0.
12EP11
(a)
dy
Show that
dx
(b)
Prove that, when
(c)
Hence find the coordinates of all points on C , for 0
2 y x cos xy
dy
dx
0, y
.
[5]
1.
[5]
x
4 , where
dy
dx
0.
[5]
Turn over
16EP13
– 11 –
M18/5/MATHL/HP1/ENG/TZ2/XX
Do not write solutions on this page.
16.
10.
[Maximum mark: 14]
f ( x) =
The function f
(a)
Find the inverse function f
(b)
,x
d
c
, stating its domain.
g ( x) =
The function g
ax + b
, for x
cx + d
[5]
2x − 3
, x∈ , x ≠ 2 .
x−2
N20/5/MATHL/HP1/ENG/TZ0/XX
B
where A , B are constants.
x−2
(i)
Express g(x) in the form A +
(ii)
Sketch the graph of y g(x) . State the equations of any asymptotes and the
8820 – 7201
13 –axes.
coordinates of any intercepts with– the
[5]
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 .
Answer all questions in the answer booklet provided. Please start each question on a new page.
[4]
10. [Maximum
[Maximum mark:
mark: 20]
11.
12]
Consider the function f x1 ax3 bx2 cx d , where x
log r x where r , x
(a) Show that log r 2 x
.
(a) (i)
Write down an2 expression for f x .
It is given
log2 ygiven
logthat
2x not
0 . exist, show that b2
(ii) that
Hence,
flog1 4does
4 x
and a , b , c , d
[2]
3ac
0.
[4]
Express y in terms of x . Give 1your
answer in the form y pxq , where p , q are
x3 3 x 2 6 x 8 , where x
Consider the function g ( 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
where
. The
area
g x x can be
written in1 the
form
p of
x R2is3 2q ,. where p , q
.
Find the value of p and the value of q .
(c) Find the value of .1
(iii) Hence find g x .
[5]
[8]
(b)
(b)
The graph of y g x may be obtained by transforming the graph of y
sequence of three transformations.
11.
.
x3 using a
(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]
[Maximum mark: 15]
Consider the curve C defined by y2
(a)
Show that
dy
y cos xy
sin xy , y
0.
12EP11
.
[5]