   Chapter 7.3, Problem 27ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
1 views

# In 26 and 27 find ( g ∘ f ) − 1 , g − 1 , f − 1 , and f − 1 ∘ g − 1 by the formaulas f ( x ) = x + 3 and g ( x ) = − x for each x ∈ R .Define f : R → R and g : R → R by the formulas f ( x ) = x + 3       and       g ( x ) = − x for each x ∈ R .

To determine

To find gof,(gof)1,g1,f1and f1og1, and explain how (g of)1 and f1o g1 are related.

Explanation

Given information:

Let X={a,c,b}, Y={x,y,z} and Z={u,v,w}. Define f:XY and g:YZ by the arrow diagrams below.

Concept used:

A function is said to be one to one function if the element in domain must be mapped with distinct

Element in codomain.

A function is onto function if element in codomain is mapped with element in domain.

Calculation:

f:RR,g:RR Defined by f(x)=x+3 and g(x)=x for all x in R.

(gf)(x)=g(f(x))=g(x+3)=x3

By definition of an inverse function, we have

(gf)1(y)= Unique real number x such that (gf)(x)=y

However, (gf)(x)=y

g(f(x))=yg(x+3)=yx3=yx=y3

(gf)1(y)=y3

Therefore, (gf)1(y)=y3

By definition of an inverse function,

g1(y)= Unique real number x, such that g(x)=y

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