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)∴
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)∴