composite functions
DESCRIPTION
trigonometryTRANSCRIPT
S.C.O. 42, Sec. 20 – C chd. 3258041, 4280095 www.gtimchd.in ; [email protected]
1
Example:
Solution :
Let f(x) = 2x 4x 3, x 3
x 4, x 3
and g(x) =
2
x 3, x 4
x 2x 2, x 4
. Describe the function
(f+g)(x) and find its domain.
2x 4x 3, x 3
x 4, 3 x 4
x 4,
f(x)
x 4
,
2
x 3, x 3
x 4, 3 x 4
x 2x
g(
2, x 4
x)
S.C.O. 42, Sec. 20 – C chd. 3258041, 4280095 www.gtimchd.in ; [email protected]
2
Example:
Solution :
Let f(x) = | x 1| ; x 1
1 x; x 1
and g(x) =
x 2; x 0
x 3; x 0
then find
(f + g)x and (fg)x.
| x 1| , x 0
| x 1| , 0 x 1
1 x, x 1
f(x)
and
x 2, x 0
x 3, 0 x 1
x 3, x 1
g(x)
| x 1|
| x 1|
x 2
x 3
, x 0
f(x) g(x) , 0
x1 x ,3
x 1
x 1
3, x 1
2x 1, 1 x 0
2x 4, 0 x 1
4, x 1
S.C.O. 42, Sec. 20 – C chd. 3258041, 4280095 www.gtimchd.in ; [email protected]
3
Let f(x) = 2x 4x 3, x 3
x 4, x 3
and g(x) =
2
x 3, x 4
x 2x 2, x 4
then find the value of
(i) (f + g)(3.5) (ii) f(g(3)) (iii) (fg)(2) (iv) (f g)(4)
Example:
Solution :
(f + g)(3.5)
= f(3.5) + g(3.5)
= (3.5 – 4) + (3.5 – 3)
= 0
f(g(3))
= f((3 – 3))
= f(0)
= (0)2 – 4(0) + 3
= 3
S.C.O. 42, Sec. 20 – C chd. 3258041, 4280095 www.gtimchd.in ; [email protected]
4
(fg)(2)
= f(2)g(2)
= ((2)2 –4(2) +3) (2 –3)
= (– 1) (– 1)
= 1
(f – g)(4)
= f(4) – g(4)
= (4 – 4) – ((4)2 + 2(4) + 2)
= – 26
Let f(x) = 2x 4x 3, x 3
x 4, x 3
and g(x) =
2
x 3, x 4
x 2x 2, x 4
S.C.O. 42, Sec. 20 – C chd. 3258041, 4280095 www.gtimchd.in ; [email protected]
5
I f the functions f(x) and g(x) are defined on R R such that
f(x) 0, x rational
x, x irrational
and g(x) =
0, x irrational
x, x rational
then (f g) (x) is (A) one-one and onto (B) neither one-one nor onto (C) one-one but not onto (D) onto but not one-one
(I I TJ EE 2005)
Example:
Solution :
(f g) : R R 0 x if x rationalf g x
x 0 if x irrational
x if x rational
x if x irrational
S.C.O. 42, Sec. 20 – C chd. 3258041, 4280095 www.gtimchd.in ; [email protected]
6
Since each branch is linear function, function is one – one
Any value from y = –x(x is rational) does not match with that of y = x(x is irrational)Also when x is rational y = –x is rational
And when x is irrational y = x is irrational
Thus function takes all real values of x
Hence function is onto
x, if x rationalf g x
x, if x irrational
S.C.O. 42, Sec. 20 – C chd. 3258041, 4280095 www.gtimchd.in ; [email protected]
Example:
Solution :
I f f(x) = nn (a x ) , x > 0, n 2, n N. Then show that (fof) (x)
= x. Find also the inverse of f(x).
f(x) = (a – xn)1/ n
f(f(x)) = (a – (f(x))n)1/n
f(f(x)) = [a – ((a – xn)1/n)n]1/n
f(f(x)) = [a – ((a – xn)]1/n
f(f(x)) = [a – a + xn]1/n
f(f(x)) = (xn)1/n
f(f(x)) = x
Since f(f(x)) = x ,
f– 1 (x) = f(x)
f– 1 (x) = (a – xn)1/n
S.C.O. 42, Sec. 20 – C chd. 3258041, 4280095 www.gtimchd.in ; [email protected]
I f f(x) = 1
1 x, then find the value of f(f(x)), f(f(f(x))),
Also find the value of n times
fofof............of(x)
Example:
Solution :
f(x) = 1
1 x fof(x) =
11 f(x)
fof(x) = 1
11
1 x
= 1 x
1 x 1
= x 1
x
S.C.O. 42, Sec. 20 – C chd. 3258041, 4280095 www.gtimchd.in ; [email protected]
9
Now fofof(x) = f(f(f(x)))
= x 1
fx
= 1x 1
1x
=
xx x 1
Finding Now n times
fofof............of(x)
Now this function depends upon the value of n.
For n = 3, 6, 9, ……. or n = 3k
n times
fofof............of(x) = x
= x
S.C.O. 42, Sec. 20 – C chd. 3258041, 4280095 www.gtimchd.in ; [email protected]
10
For n = 3k + 1 ( n = 1, 4, 7, ………)
n times
fofof............of(x) = 1
1 x
For n = 3k + 2 ( n = 2, 5, 8, ………)
n times
fofof............of(x) = x 1
x
S.C.O. 42, Sec. 20 – C chd. 3258041, 4280095 www.gtimchd.in ; [email protected]
11
Example:
Solution :
Let f(x) = ax + b and g(x) = cx + d, a 0, c 0. Assume a = 1, b = 2. If (fog) (x) = (gof) (x) for all x, what can you say about c and d ?
(fog) (x) = f(g(x)) = a(cx + d) + b
(gof) (x) = f(f(x)) = c(ax + b) + d
Given that, (fog) (x) = (gof) (x) and at a = 1, b = 2
cx + d + 2 = cx + 2c + d
Comparing constant term, d + 2 = 2c + d
c = 1 and d is arbitrary
S.C.O. 42, Sec. 20 – C chd. 3258041, 4280095 www.gtimchd.in ; [email protected]
Let f(x) = x
,x 1.x 1
Then for what value of is f(f(x)) = x?
(A) 2 (B) 2 (C) 1 (D) – 1(I I TJ EE 2001)
Example:
Solution :
xf x ,
x 1
x 1
ff x x 2x
x1 x 1
xx 1
xx
1x 1
2 21 x 1 x 0
This is an identity in x
= 1
+ 1 = 0 and 1 2 = 0
S.C.O. 42, Sec. 20 – C chd. 3258041, 4280095 www.gtimchd.in ; [email protected]
13
Example:
Solution :
I f f be the greatest integer function and g be the modulus
function, then find the value of (gof)53
(fog)53
f(x) = [x] and g(x) = |x|5 5
(gof) (fog)3 3
= 5 5
g ff g3 3
= 5 5
g f3 3
= 5g 2 f
3
= 2 – 1 = 5
| 2 |3
= 1
S.C.O. 42, Sec. 20 – C chd. 3258041, 4280095 www.gtimchd.in ; [email protected]
14
Let f(x) =1 | x | , x 1
[x], x 1
, where [.] denotethe
greatest integer function. Then find the value of f(f( 2.3))
Example:
Solution : f(x) = 1 | x | , x 1
[x], x 1
.
First find f( 2.3)
For x < –1, f(x) = 1 + |x|
f( 2.3) = 1 + | 2.3|
f( 2.3) = 3.3
Now f(f( 2.3)) = f(3.3)
For x –1, f(x) = [x]
f(f(2.3)) = f(3.3) = [3.3] = 3
S.C.O. 42, Sec. 20 – C chd. 3258041, 4280095 www.gtimchd.in ; [email protected]
15
I f g(x) = x2 + x 2 and 2gof(x) 4x 10x 4 , then find f(x)
Example:
Solution :2g(f(x)) 4x 10x 4 ……(i)
Also from g(x) = x2 + x 2
g(f(x)) = (f(x))2 + f(x) – 2 ……(ii)
Comparing (i) and (ii)
(f(x))2 + f(x) 2 = 4x2 10x + 4
f(x)2 + f(x) (4x2 – 10x + 6) = 0
f(x) = 21 1 4(4x 10x 6)
2
21 16x 40x 252
S.C.O. 42, Sec. 20 – C chd. 3258041, 4280095 www.gtimchd.in ; [email protected]
16
1 (4x 5)2
= 2x – 3 or 2 – 2x
S.C.O. 42, Sec. 20 – C chd. 3258041, 4280095 www.gtimchd.in ; [email protected]
17
Suppose that g(x) = 1 + x and f(g(x)) = 3 + 2 x + x, then find function f(x)
Example:
Solution :
g(x) = 1 + x and f(g(x)) = 3 + 2 x + x …….(i)
f(1 + x ) = 3 + 2 x + x
Put 1 + x = y x = (y 1)2 f(y) = 3 + 2(y 1) + (y 1)2
= 2 + y2
f(x) = 2 + x2
S.C.O. 42, Sec. 20 – C chd. 3258041, 4280095 www.gtimchd.in ; [email protected]
18
Example:
Solution :
I f f(x) = sin x + cos x, g(x) = x2 1, then g(f(x)) is invertible in the domain
(A) 0,2
(B) ,
4 4
(C) ,2 2
(D)[0,]
(I I TJ EE 2004)
f(x) = sin x + cos x, g(x) = x2 1
g(f(x)) = (sin x + cos x)2 1
= sin2x + cos2x + 2sin x cos x – 1 = sin 2x
S.C.O. 42, Sec. 20 – C chd. 3258041, 4280095 www.gtimchd.in ; [email protected]
19
x4 4
We know that sin is invertible when /2 /2
g(f(x)) is invertible in 2x2 2