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Chapter 2: Q 93. (page 92)

Can a function be both even and odd? Explain.

Short Answer

Expert verified

The function can be both even and odd if f(x)=0for all x.

Step by step solution

01

Step 1. Define even and odd function

A function is odd if f(-x)=-f(x)…(1)and a function is even if f(-x)=f(x)…(2)

02

Step 2. Explain when the function be both even and odd.

From the eqution (2), the value of f(x)is as follows:

f(x)=-f(-x)…(3)

Add equation (1)and (3).

f(x)+f(x)=f(-x)+(-f(-x))2f(x)=f(-x)-f(-x)2f(x)=0⇒f(x)=0

Hence, the function can be both even and odd if f(x)=0for all x.

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True or False

To obtain the graph of f(x)=x+2, shift the graph ofy=x horizontally to the right 2units.

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To obtain the graph of f(x)=x3+5,shift the graph of y=x3vertically up 5units.

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