Question 3:
Given that f : x → hx + k and f2 : x → 4x + 15.
(a) Find the value of h and of k.
(b) Take the value of h > 0, find the values of x for which f (x2 ) = 7x
Solution:
(a)
Given f (x) = hx + k
f2 (x) = ff (x) = f (hx + k)
= h (hx + k) + k
= h2 x + hk + k
f2 (x) = 4x + 15
h2 x + hk + k = 4x + 15
h2 = 4
h = ± 2
when, h = 2
hk + k = 15
2k + k = 15
k = 5
When, h = –2
hk + k = 15
–2k + k = 15
k = –15
(b)
h > 0, h = 2, k = 5
Given f (x) = hx + k
f (x) = 2x + 5 f (x2 ) = 7x
2 (x2 ) + 5 = 7x
2x2 – 7x + 5 = 0
(2x – 5)(x –1) = 0
2x – 5 = 0 or x –1= 0
x = 5/2 x = 1
Given that f : x → hx + k and f2 : x → 4x + 15.
(a) Find the value of h and of k.
(b) Take the value of h > 0, find the values of x for which f (x2 ) = 7x
Solution:
(a)
Given f (x) = hx + k
f2 (x) = ff (x) = f (hx + k)
= h (hx + k) + k
= h2 x + hk + k
f2 (x) = 4x + 15
h2 x + hk + k = 4x + 15
h2 = 4
h = ± 2
when, h = 2
hk + k = 15
2k + k = 15
k = 5
When, h = –2
hk + k = 15
–2k + k = 15
k = –15
(b)
h > 0, h = 2, k = 5
Given f (x) = hx + k
f (x) = 2x + 5 f (x2 ) = 7x
2 (x2 ) + 5 = 7x
2x2 – 7x + 5 = 0
(2x – 5)(x –1) = 0
2x – 5 = 0 or x –1= 0
x = 5/2 x = 1
Question 4:
The function f is denoted by f:x→1+x1−x,x≠1. Find f2,f3,f4 and hence write down the functions f51 and f52.
Solution:
f(x)=1+x1−x,x≠1f2(x)=f[f(x)]=f(1+x1−x) =1+(1+x1−x)1−(1+x1−x)=1−x+1+x1−x1−x−1−x1−x =2−2x=−1xf3(x)=f[f2(x)]=f(−1x) =1+(−1x)1−(−1x)=x−1xx+1x =x−1x+1f4(x)=f[f3(x)]=f(x−1x+1) =1+(x−1x+1)1−(x−1x+1)=x+1+x−1x+1x+1−x+1x+1 =2x2=xf5(x)=f[f4(x)]=f(x)=1+x1−x(recurring)∴f51(x)=f3[f48(x)]=f3(x) =x−1x+1f52(x)=f4[f48(x)]=f4(x)=x
The function f is denoted by f:x→1+x1−x,x≠1. Find f2,f3,f4 and hence write down the functions f51 and f52.
Solution:
f(x)=1+x1−x,x≠1f2(x)=f[f(x)]=f(1+x1−x) =1+(1+x1−x)1−(1+x1−x)=1−x+1+x1−x1−x−1−x1−x =2−2x=−1xf3(x)=f[f2(x)]=f(−1x) =1+(−1x)1−(−1x)=x−1xx+1x =x−1x+1f4(x)=f[f3(x)]=f(x−1x+1) =1+(x−1x+1)1−(x−1x+1)=x+1+x−1x+1x+1−x+1x+1 =2x2=xf5(x)=f[f4(x)]=f(x)=1+x1−x(recurring)∴f51(x)=f3[f48(x)]=f3(x) =x−1x+1f52(x)=f4[f48(x)]=f4(x)=x