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Chapter 5: Q 45. (page 257)

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

f(x)=2x;g(x)=12x

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)=2xg(x)=12x

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 substituteg(x)=12xin the function f(g(x)),

Then the function will becomerole="math" localid="1646295891765" f(12x).

Now replace x with 12xin role="math" localid="1646295985713" f(x)=2x,

f(12x)=2(12x)=x

Therefore,(f鈭�g)(x)=x

03

Step 3. Find (g∘f)(x).

(g鈭�f)(x)=g(f(x))

Substitute2xin the functiong(f(x)).

Then the function will become g(2x),

g(2x)=12(2x)=x

So, (g鈭�f)(x)=x

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

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