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Chapter 5: Q 52. (page 257)

In Problems 45鈥�52, show that (f鈭�g)(x)=(g鈭�f)(x)=x

f(x)=1x;g(x)=1x

Short Answer

Expert verified

(f鈭�g)(x)=x(g鈭�f)(x)=x

Therefore,(f鈭�g)(x)=(g鈭�f)(x)=x.

Step by step solution

01

Step 1. Given information.

The given composite function is:

f(x)=1xg(x)=1x

When we are given two functions f and g, the composite function which is denoted byf鈭�g is defined by (f鈭�g)(x)=f(g(x)).

02

Step 2. Find (f∘g)(x).

(f鈭�g)(x)=f(g(x))

Now substituteg(x)=1x in the function f(g(x)),

Then the function will become f(1x).

f(1x)=11x=x

Therefore,(f鈭�g)(x)=x.

03

Step 3. Find (g∘f)(x).

(g鈭�f)(x)=g(f(x))

Substitutef(x)=1xin the functionrole="math" localid="1646310351413" g(f(x)),

role="math" localid="1646309963204" (g鈭�f)(x)=g(f(x))=g(1x)=11x=x

Therefore, (g鈭�f)(x)=x.

It is shown that(f鈭�g)(x)=(g鈭�f)(x)=x.

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