Question

let,f,g.h are functions from R to Rdefined as f(x)=x+2,g(x)=1/(x^2+1),h(x)=3 .compute f^(-1).g(x) and h.f.(g.f^(-1)).(h.f(x))

Answer 1

Answer:

(i)

To prove:

(f+g)oh=foh+goh

Consider:

((f+g)oh)(x)

=(f+g)(h(x))

=f(h(x))+g(h(x))

=(foh)(x)+(goh)(x)

=(foh)+(goh)(x)

∴((f+g)oh)(x)=(foh)+(goh)(x)    for all x∈R

Hence, (f+g)oh=foh+goh

(ii)

To prove: (f.g)oh=(foh).(goh)

Consider: 

((f.g)oh)(x)

=(f.g)(h(x))

=f(h(x)).g(h(x))

=(foh)(x).(goh)(x)

=(foh).(goh)(x)

∴((f.g)oh)(x)=(foh).(goh)(x) for allx∈R

Hence, (f.g)oh=(foh).(goh)

Video Explanation