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Chapter 5: Q 46. (page 257)

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

f(x)=4x;g(x)=14x

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)=4xg(x)=14x

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

02

Step 2. Find (f∘g)(x).

(f鈭�g)(x)=f(g(x))

Now substitute g(x)=14xin the function f(g(x)),

(f鈭�g)(x)=f(g(x))=f(14x)=4(14x)=x

Thus,(f鈭�g)=x.

03

Step 3. Find (g∘f)(x)

(g鈭�f)(x)=g(f(x))

Substitutef(x)=4xin the functionf(g(x)),

(g鈭�f)(x)=g(f(x))=g(4x)=14(4x)=x

Thus, (g鈭�f)(x)=x.

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

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