Ex 1.5 | Aster Classes

Question 1.


Using the functions f and g given below, find fog and gof Check whether fog = gof.

(i) f(x) = x – 6, g(x) = x2
Answer:
f(x) = x – 6, g(x) = x2
fog = fog (x)
= f(g(x))
fog = f(x)2
= x2 – 6
gof = go f(x)
= g(x – 6)
= (x – 6)2
= x2 – 12x + 36
fog ≠ gof


(ii) f(x) = 2x, g(x) = 2x2 – 1
Answer:
f(x) – 2x; g(x) = 2x2 – 1
fag = f[g (x)]
= f(2x2 – 1)
= 22×2−1
gof = g [f(x)]
= g (2x)
= 2 (2x)2 – 1
=2×4×2−1
=8×2−1
fog ≠ gof

(iii) f(x) = x+63, g(x) = 3 – x
Answer:
f(x) = x+6x, g(x) = 3 – x
fog = f[g(x)]
= f(3 – x)


(iv) f(x) = 3 + x, g(x) = x – 4
Answer:
f(x) = 3 + x ;g(x) = x – 4
fog = f[g(x)]
= f(x – 4)
= 3 + x – 4
= x – 1
gof = g[f(x)]
= g(3 + x)
= 3 + x – 4
= x – 1
fog = gof

(v) f(x) = 4x2 – 1,g(x) = 1 + x
Answer:
f(x) = 4x2 – 1 ; g(x) = 1 + x
fog = f[g(x)]
= 4(1 + x)
= 4(1 + x)2 – 1
= 4[1 + x2 + 2x] – 1
= 4 + 4x2 + 8x – 1
= 4x2 + 8x + 3
gof = g [f(x)]
= g (4x2 – 1)
= 1 + 4x2 – 1
= 4x2
fog ≠ gof


Question 2.
Find the value of k, such that fog = gof
(i) f(x) = 3x + 2, g(x) = 6x – k
(ii) f(x) = 2x – k, g(x) = 4x + 5
Solution:
(i) f(x) = 3x + 2, g(x) = 6x – k
fog(x) = f(g(x)) = f(6x – k) = 3(6x – k) + 2
= 18x – 3k + 2 …………… (1)
gof(x) = g(f(x)) = g(3x + 2) = 6(3x + 2) – k
= 18x + 12 – k ……………. (2)
(1) = (2)
⇒ 18x – 3k + 2 = 18x + 12 – k
2k = -10
k = -5

(ii) f(x) = 2x – k, g(x) = 4x + 5
fog(x) = f(g(x)) = f(4x + 5) = 2(4x + 5) – k
= 8x + 10 – k ……………… (1)
gof(x) = g(f(x)) = g(2x – k) = 4(2x – k) + 5
= 8x – 4k + 5 ……………. (2)
(1) = (2)
⇒ 8x + 10 – k = 8x – 4k + 5
3k = -5
k = −53


Question 3.
If f(x) = 2x – 1, g(x) = x+12, show that f o g = g o f = x
Answer:
f(x) = 2x – 1 ; g(x) = x+12
fog = f[g(x)]

∴ fog = gof = x
Hence it is proved.


Question 4.
(i) If f (x) = x2 – 1, g(x) = x – 2 find a, if gof(a) = 1.
(ii) Find k, if f(k) = 2k – 1 and fof (k) = 5.
Solution:
(i) f(x) = x2 – 1, g(x) = x – 2
Given gof(a) = 1
gof(x) = g(f(x)
= g(x2 – 1) = x2 – 1 – 2
= x2 – 3
gof(a) ⇒ a2 – 3 = 1 =+ a2 = 4
a = ± 2
(ii) f(k) = 2k – 1
fo f(k) = 5
f(f(k)m = f(2k – 1) = 5
⇒ 2(2k – 1) – 1 = 5
4 k – 2 – 1 = 5 ⇒ 4k = 8
k = 2


Question 5.
Let A,B,C ⊂ N and a function f: A → B be defined by f(x) = 2x + 1 and g: B → C be defined by g(x) = x2 . Find the range of fog and gof.
Answer:
f(x) = 2x + 1 ; g(x) = x2
fog = f[g(x)]
= f(x2)
= 2x2 + 1
2x2 + 1 ∈ N
g o f = g [f(x)]
= g (2x + 1)
g o f = (2x + 1)2
(2x + 1)2 ∈ N
Range = {y/y = 2x2 + 1, x ∈ N};
{y/y = (2x + 1)2, x ∈ N)


Question 6.
If f(x) = x2 – 1. Find (i)f(x) = x2 – 1, (ii)fofof
Solution:
(i) f(x) = x2 – 1
fof(x) = f(fx)) = f(x2 – 1)
= (x2 – 1 )2 – 1;
= x4 – 2x2 + 1 – 1
= x4 – 2x2
(ii) fofof = f o f(f(x))
= f o f (x4 – 2x2)
= f(f(x4 – 2x2))
= (x4 – 2x2)2 – 1
= x8 – 4x6 + 4x4 – 1


Question 7.
If f : R → R and g : R → R are defined by f(x) = x5 and g(x) = x4 then check if f, g are one – one and fog is one – one?
Answer:
f(x) = x5 – It is one – one function
g(x) = x4 – It is one – one function
fog = f[g(x)]
= f(x4)
= (x4)5
fag = x20
It is also one-one function.


Question 8.
Consider the functions f(x), g(x), h(x) as given below. Show that (fog)oh = fo(goh) in each case.
(i) f(x) = x – 1, g(x) = 3x + 1 and h(x) = x2
(ii) f(x) = x2, g(x) = 2x and h(x) = x + 4
(iii) f(x) = x – 4, g(x) = x2 and h(x) = 3x – 5
Solution:
(i) f(x) = x – 1, g(x) = 3x + 1 and h(x) = x2
f(x) = x – 1
g(x) = 3x + 1
f(x) = x2
(fog)oh = fo(goh)
LHS = (fog)oh
fog = f(g(x)) = f(3x + 1) = 3x + 1 – 1 = 3x
(fog)oh = (fog)(h(x)) = (fog)(x) = 3……………. (1)
RHS = fo(goh)
goh = g(h(x)) = g(x2) = 3x2 + 1
fo(goh) = f(3x2 + 1) = 3x2 + 1 – 1= 3x………… (2)
LHS = RHS Hence it is verified.

(ii) f(x) = x2, g(x) = 2x, h(x) = x + 4
(fog)oh = fo(goh)
LHS = (fog)oh
fog = f(g(x)) = f(2x) = (2x)2 = 4x2
(fog)oh = (fog) h(x) = (fog) (x + 4)
= 4(x + 4)2 = 4(x2 + 8x+16)
= 4x2 + 32x + 64 ………….. (1)
RHS = fo(goh) goh = g(h(x)) = g(x + 4)
= 2(x + 4) = (2x + 8)
fo(goh) = f(goh) = f(2x + 8) = (2x + 8)2
= 4x2 + 32x + 64 ……………… (2)
(1) = (2)
LHS = RHS
∴ (fog)oh = fo(goh) It is proved.

(iii) f(x) = x – 4, g(x) = x2, h(x) = 3x – 5
(fog)oh = fo(goh)
LHS = (fog)oh
fog = f(g(x)) = f(x2) = x2 – 4
(fog)oh = (fog)(3x – 5) = (3x – 5)2 – 4
= 9x2 – 30x + 25 -4
= 9x2 – 30x + 21 ………….. (1)
∴ RHS = fo(goh)
(goh) = g(h(x)) = g(3x – 5) = (3x – 5)2
= 9x2 – 30x + 25
fo(goh) = f(9x2 – 30 x + 25)
= 9x2 – 30x + 25 – 4
= 9x2 – 30x + 21 …………… (2)
(1) = (2)
LHS = RHS
∴ (fog)oh = fo(goh)
It is proved.


Question 9.
Let f = {(-1, 3), (0, -1), (2, -9)} be a linear function from Z into Z. Find f(x).
Answer:
The linear equation is f(x) = ax + b
f(-1) = 3
a(-1) + b = 3
-a + b = 3 ….(1)
f(0) = -1
a(0) + b = -1
0 + b = -1
b = -1
Substitute the value of b = -1 in (1)
-a – 1 = 3
-a = 3 + 1
-a = 4
a = -4
∴ The linear equation is -4(x) -1 = -4x – 1 (or) – (4x + 1)


Question 10.
In electrical circuit theory, a circuit C(t) is called a linear circuit if it satisfies the superposition principle given by C(at1 + bt2) = aC(t1) + bC(t2), where a,b are constants. Show that the circuit C(t) = 31 is linear.
Solution:
Given C(t) = 3t. To prove that the function is linear
C(at1) = 3a(t1)
C(bt2) = 3 b(t2)
C(at1 + bt2) = 3 [at1 + bt2] = 3at1 + 3bt2
= a(3t1) + b(3t2) = a[C(t1) + b(Ct2)]
∴ Superposition principle is satisfied.
Hence C(t) = 3t is linear function.


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