To determine (1) If f and g are both even functions,

Anonym

Anonym

Answered question

2021-10-12

To determine
(1) If f and g are both even functions, whether f+g is even?
(2) If f and g are both odd functions, whether f+g is odd?
(3) If f is even and g is odd function, what will f+g?

Answer & Explanation

sweererlirumeX

sweererlirumeX

Skilled2021-10-13Added 91 answers

A function is even when
f(x)=f(x)
for all x.
A functon is odd when
f(x)=f(x)
for all x.
Consider (f+g)(x),
(f+g)(x)=f(x)+g(x)
=f(x)+g(x) (f and g are even functions)
=(f+g)(x)
So, f+g is also even function.
f is odd so, f(x)=f(x), for all x.
g is odd so, g(x)=g(x), for all x.
Consder (f+g)(x),
(f+g)(x)=f(x)+g(x)
=f(x)+g(x)
(g(x)) (f and g are odd function)
=f(x)g(x)
=(f(x)+g(x))
=(f+g)(x)
So, f+g is also odd fuction.
f is an even function so, f(x)=f(x), for all x.
g is an odd function so, g(x)=g(x), for all x.
Let f(x)=x2
Since, f(x)=(x)2=x2
So, f(x)=x2 is an even function.
Let g(x)=x3
Since, g(x)=(x)3=x3
So, g(x)=x3 is an odd function.
Consider (f+g)(x),
(f+g)(x)=f(x)+g(x)
=(x)2+(x)3
=x2x3
and (f+g)(x)=f(x)+g(x)
=x2+x3
We observe that (f+g)(x)(f+g)(x) and (f+g)(x)(f+g)(x).
So, the sum of an even and an odd function is neither even non odd.

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