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Chapter 5: Q 50. (page 257)

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

f(x)=4-3x;g(x)=13(4-x)

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)=4-3xg(x)=13(4-x)

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)=13(4-x)in the function f(g(x)),

Then the function will become f(13(4-x)).

f(13(4-x))=4-3(13(4-x))=4-3(4-x3)=4-4+x=x

Therefore,(f鈭�g)(x)=x.

03

Step 3. Find (g∘f)(x).

(g鈭�f)(x)=g(f(x))

Substitutef(x)=4-3xin the functiong(f(x)),

localid="1646305216024" (g鈭�f)(x)=g(f(x))=g(4-3x)=13(4-(4-3x))=13(4-4+3x)=x

Therefore, (g鈭�f)(x)=x.

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

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