Chapter 1.4, Problem 53E

Solution

a.

To find : Find the identity function for the function $g\left(x\right)$ for a ddition. let $f\left(x\right),g\left(x\right)$ be the function then function then $\left(f+g\right)\left(x\right)=\left(g+f\right)\left(x\right)=f(x)$

For the identity function for the function $g\left(x\right)=I_d\left(x\right)=0$

Given information:

let $f\left(x\right),g\left(x\right)$ be the function then function and $\left(f+g\right)\left(x\right)=\left(g+f\right)\left(x\right)=f\left(x\right)$ .

Concept used:

An identity for a function operation is a function that combines with a given function $f$ to return the same function $f$ .

Calculations:

Suppose take identity function for $g\left(x\right)=$

  $I_d\left(x\right)=0,0\in R$ and let $f\left(x\right)$ be the function.

Then

  $\begin{array}{c}\left(f+g\right)\left(x\right)=f\left(x\right)+g\left(x\right)\\ =f\left(x\right)+0\\ =f\left(x\right)\\ \left(g+f\right)\left(x\right)=g\left(x\right)+f\left(x\right)\\ =0+f\left(x\right)\\ =f\left(x\right)\end{array}$

So for the identity function for the function $g\left(x\right)=I_d\left(x\right)=0$

b.

To find : Find the identity function for the function $g\left(x\right)$ for multiplication . let $f\left(x\right),g\left(x\right)$ be the function then function then $\left(fg\right)\left(x\right)=\left(gf\right)\left(x\right)=f(x)$

For the identity function for the function $g\left(x\right)=I_d\left(x\right)=1$

Given information:

let $f\left(x\right),g\left(x\right)$ be the function then function and $\left(fg\right)\left(x\right)=\left(gf\right)\left(x\right)=f\left(x\right)$ .

Concept used:

An identity for a function operation is a function that combines with a given function $f$ to return the same function $f$ .

Calculations:

Suppose take identity function for $g\left(x\right)=$

  $I_d\left(x\right)=1,1\in R$ and let $f\left(x\right)$ be the function.

Then

  $\begin{array}{c}\left(fg\right)\left(x\right)=f\left(x\right)\times g\left(x\right)\\ =f\left(x\right)\times 1\\ =f\left(x\right)\\ \left(gf\right)\left(x\right)=g\left(x\right)\times f\left(x\right)\\ =1\times f\left(x\right)\\ =f\left(x\right)\end{array}$

So for the identity function for the function $g\left(x\right)=I_d\left(x\right)=1$

c.

To find : Find the identity function for the function $g\left(x\right)$ for multiplication . let $f\left(x\right),g\left(x\right)$ be the function then function then $\left(f^{0}g\right)\left(x\right)=\left(g^{o}f\right)\left(x\right)=f(x)$

For the identity function for the function $g\left(x\right)=I_d\left(x\right)=x$

Given information:

let $f\left(x\right),g\left(x\right)$ be the function then function and $\left(f^{0}g\right)\left(x\right)=\left(g^{0}f\right)\left(x\right)=f\left(x\right)$ .

Concept used:

An identity for a function operation is a function that combines with a given function $f$ to return the same function $f$ .

Calculations:

Suppose take identity function for $g\left(x\right)=$

  $I_d\left(x\right)=x$ and let $f\left(x\right)$ be the function.

Then

  $\begin{array}{c}\left(f^{0}g\right)\left(x\right)=f\left(g\left(x\right)\right)\\ =f\left(x\right)\\ =g\left(f\left(x\right)\right)\\ =g^{o}f\left(x\right)\end{array}$

So for the identity function for the function $g\left(x\right)=I_d\left(x\right)=x$