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Chapter 5: Q 48. (page 257)

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

f(x)=x+5;g(x)=x-5

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)=x+5g(x)=x-5

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)=x-5in the function f(g(x)),

Then the function will become f(x-5),

f(x-5)=x-5+5=x

Therefore,(f鈭�g)(x)=x.

03

Step 3. Find (g∘f)(x).

(g鈭�f)(x)=g(f(x))

Substitutef(x)=x+5in the functiong(f(x))

(g鈭�f)(x)=g(f(x))=g(x+5)=x+5-5=x

Therefore,(g鈭�f)(x)=x.

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

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Most popular questions from this chapter

Evaluate each expression using the graphs of y=f(x)and y=g(x)shown in the figure.

(a)g鈭�f-1 (b) g鈭�f0(c) f鈭�g-1(d) f鈭�g4.

If fx=ax, show thatf-x=1fx.

If f(x)=ax+band g(x)=cx+d, then find: -

(a)f鈭�g

(b)g鈭�f

(c)the domain of f鈭�gand g鈭�f

(d)the conditions for whichf鈭�g=g鈭�f

Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red.

True or False

To graph (x-2)3, shift the graph of y=x3to the left 2 units.

A function y=f(x)is decreasing on the interval(0,5).What conclusions can you draw about the graph ofy=f-1(x)?
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