0% found this document useful (0 votes)
14 views10 pagesFunctions HW
The document contains a series of mathematical problems involving functions f and g, including finding ranges, expressions for inverse functions, and solving equations. It also discusses the conditions for one-one functions and the composition of functions. Various values of constants and domains are explored throughout the problems.
Uploaded by
ammarkalwar2007Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF or read online on Scribd
0% found this document useful (0 votes)
14 views10 pagesFunctions HW
The document contains a series of mathematical problems involving functions f and g, including finding ranges, expressions for inverse functions, and solving equations. It also discusses the conditions for one-one functions and the composition of functions. Various values of constants and domains are explored throughout the problems.
Uploaded by
ammarkalwar2007Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF or read online on Scribd
/ 10
9, Mid 21/P11/09
Functions f and g are defined as follows:
f(x) = (x2)? =4 forx 22,
g(x) =ax+2 forxeR,
where a is a constant.
(a) State the range of f.
(b) Find f-!(x).
(©) Given that a = -, solve the equation f(x) = g(x).
(@) Given instead that ggf~!(12) = 62, find the possible values of a.
@ fas 2-4 x v2 (4) Fi) (x-2)?-4
= (4-2)74
@-2)4 ae 2
fa) = -4 2 Eyes a
f'@): 22h
fal>-4 @
(a
(fads (x-2)?-4
g(a) = an4 2.
a-s
3
GOs ape
2
ca
(h-aYous 5 x42
3
Wythe - 5 42
3
stents ala led
3
®-D4-2:0
Ne3BS Ws 2 x
3
Made with Goodnotes:
a
(2)
B)
(5) 13. OIN 20/P11/Q9
The functions f and g are defined by
f(x) =27 +3 forx>0,
g(x) =2x+1 forx>-}.
(a) Find an expression for fg(x)..
u)
(b) Find an expression for (fg)"!(x) and state the domain of (fg)! (4)
(©) Solve the equation fg(x) - 3 = gf(x)- 14]
@ Fod= x43 (9 tya)= emtas — £Q= coud?
g(d= 2x41
FQ)= (2xti?43
fq')--1 a hes
2
15. O/N 20/P13/Q6
‘The function f is defined by f(x) = ==> for x > 4
(a) Find an expression for f~!(x).
Qi?
(©) Show that 5 + 5755
(c) State the range of f.
can be expressed as
mye
2a
gis 202341
2a? +64]
gf @= 2987
fq) -3= gf (a)
(0 ati) iz 22 = 2047
Waites 2007
Whr21-6= 0
BI
2)
a
2x
@ fad= 2x () 242 QO f@)= 20
34-1 = 3G 3x1
Wa | 2x 23x) to man= 2.
Bat 3x) 3
Say-y = 20 bx- 249 Ff) > 2
3x1) 3
Bays DK ely Ma > Ba
4(34-2) =4 1362-) 3x4
Pane
3y-2
Plays
na
Made with Goodnotes: 18, M/J 20/P13/Q9
‘The functions f and g are defined by
f(x) =27 -4x+3 forx>c, where
LL forx>-1.
xe
(a) Express f(x) in the form (x- a)? +b.
g(x) =
Itis given that f is a one-one function. Q)
(b) State the smallest possible value of c.
It is now given that c= 5.,
uy
(©) Find an expression for f~'(x) and state the domain of f~'. B]
(@) Find an expression for gf(x) and state the range of gf. B)
@ fla) = euro (u) fed f-2)*)
(aft 2 a2 aan sett -2)5 |
2osk atbes 1-2 NT
Q=2 2%bbes x >20d7
| bet
2)
© ¢aye-2)-1 (0) fo -2)%-1
(n=2)* go) = L.
aH
4@) 2
6-2)
Fa)=_1
2 O-2)"
Made with Goodnotes: 19. MAR 20/P12/Q9
(a) Express 2x? + 12x + 11 in the form 2(x + a)* + b, where a and b are constants.
The function f is defined by f(x) = 2x7 + 12x + 11 for.x < -4.
(b) Find an expression for f~!(x) and state the domain of f~!.
The function g is defined by g(x) = 2x—3 forx < k.
(©) For the case where k = —1, solve the equation fg(x) = 193.
(d) State the largest value of k possible for the composition fg to be defined.
@ 2p 2x0 (6) f@s 2005%7
Liaray te => ax™har +20} Ya? = Ge3*
hasia 2d 2
ae3 p= Ip {Ma7) |= x49
2 (ut3)?-7 2
K= -3 el
© fade 2quy%7 *ar
ga)- 20-3 fa): -2+ [ao
fyla)= 193 2
£Q)= 2C(ax-92 D7 flaje -3- [oar
fi) 2(24)*-7 2
tye) = 390%
aa-1= 193 @ 24-3 6-4
es 25 das -l
ae tS be le [ey
es 2
Vnagest wat et sit a — L
bi
Made with Goodnotes:
2]
G3)
[2]
0) 30. O/N 18/P11/Q11
(a) The one-one function f is defined by f(x) = (x- 3)’ - 1 for x 0.
(Show that gg(2x) can be expressed in the form (2x-3)* + b(2x— 3)? +c, where b and c are
constants to be found.
QI)
(ii) Hence expand gg(2x) completely, simplifying your answer. [4]
Oi fee wg%
(a-9* <1
x > 321
rr
i, FO) = Gee
ytle (7-8)*
Eiger + x-8
3 tlie
fhe 3
Made with Goodnotes: 34. M/J 18/P13/Q10
‘The one-one function f is defined by f(x) = (x- 2)? + ag
g
(@® State the smallest possible value of c. @ ia]
In parts (ii) and (iii) the value of c is 4.
(ii) Find an expression for f~!(x) and state the domain of f~!. B)
Solve the equation ff(x) = 51, giving your answer in the form a + vb. [5]
0) LUD
it) fo): G-2)?s2 Range ot -2)*
ti) tao Goh igs fe) ee *42
@= Prar2 HA) > 6
(i) fa): @-297s2
1): a-a)a-2)*
$lod= (A 4x44)”
a I6x"4 lo =S)
Tox 3520
Pu
ur Iby- 3520
Us 9437 us 3-300
Ws 143d 7 ae Bah
te 247
Made with Goodnotes: 64, O/N 12/P13/Q7
(i) The diagram shows part of the curve y = 11 - x?
and part of the straight line y = 5 - x meeting at
the point A (p, q), where p and q
are positive constants. Find the values of p and q,
yel-x?
B]
(ii) The function f is defined for the domain x > 0 by
f(x) ={n" for0p.
tetas
Express f-!(x) in a similar way. 15] AGO
Made with Goodnotes: 65. M/J 12/P12/Q10
Functions f and g are defined by
@ Obtain expressions, in terms of x, for f-'(x) and g™!(x),
is not defined.
(ii) Sketch the graphs of y = f(x) and y = f~!(x) on the
between the two graphs.
(ili) Given that the equation fg(x) = 5 - kx, where
possible values of k.
9o)= 2
a
gs 8.
3
deseo
y= 5 48
g
(i) $@) = Das
= 8.
a
£(g)=
See
Fg) = 16
5 48
de 45 = Sku
HD
de 2 - hx
ed
I = ~kx (x-3)
lem -hat+ Bh
Kx? -3kx+lb= 0
bac