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Chapter 5: Q. 77 (page 258)

If f is an odd function and gis an even function, show that the composite functionsf鈭�g and g鈭�f are both even.

Short Answer

Expert verified

The, f鈭�gand g鈭�fboth are even function.

Step by step solution

01

Step 1. Given Information

Given that fis odd function andgis even function.

02

Step 2. Solution

A function fis said to be odd if f(-x)=-f(x)鈭赌x.

A function gis said to be even if role="math" localid="1646315555532" g(-x)=g(x)鈭赌x.

Since

f鈭�g(x)=f(g(x))

Now,

f鈭�g(-x)=f(g(-x))f鈭�g(-x)=f(g(x)){gisevenfunction.}f鈭�g(-x)=f鈭�g(x)鈭赌x

So, f鈭�gis an even function.

For g鈭�f(x)=g(f(x)).

g鈭�f(-x)=g(f(-x))g鈭�f(-x)=g(-f(x)){fisoddfunction}g鈭�f(-x)=g(f(x)){gisevenfunction}g鈭�f(-x)=g鈭�f(x)鈭赌x

Sog鈭�fis an even function.

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