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Chapter 5: Q 49. (page 257)

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

f(x)=2x-6;g(x)=12(x+6)

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)=2x-6g(x)=12(x+6)

When we are given two functions f and g, the composite function which is denoted byf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)=12(x+6)in the function f(g(x)),

Then the function will become f(12(x+6)),

f(12(x+6))=2(12(x+6))-6=2(x2+3)-6=2x2+6-6=x

Therefore,(f鈭�g)(x)=x.

03

Step 3. Find (g∘f)(x).

(g鈭�f)(x)=g(f(x))

Substitutef(x)=2x-6in the functiong(f(x)),

localid="1646304263647" (g鈭�f)(x)=g(f(x))=g(2x-6)=12(2x-6+6)=12(2x)=x

Therefore, (g鈭�f)(x)=x.

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

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