Exercise 123 E and Odd Functions A function f is said to be

Exercise 1.2.3. (E and Odd Functions) A function f is said to be even if f(-r) ven f(ar) and odd if f (-r) f(r) for every a in its domain. For erample, f(r) r2, g(a) COST are even function whereas the functions r (r) 333, s(r) sina are odd functions. For any two functions f and g answer the following questions. (a) What can be said about f+ g? Even, odd, or not necessarily either in each of the following cases? (ij both f and g are even functions. (ii) f is even and g is odd. (ii, both f and g are odd functions (i) f is odd and g is even. (b) Do the same for f.g.

Solution

1 . If f , g are even functions then f + g is an even function . because ( f + g) ( - x ) = f ( - x ) + g ( - x)

= f ( x ) + g ( x ) as both f , g ARE EVEN

2 . if f , g are odd hen ( f + g )( - x) = f ( - x ) + g ( - x ) = - f ( X) - g ( x) = - (f + g ) ( x)

.hence f+ g is also odd

3 . If f is even and g is odd then f + g need not be even or neednot be odd

example f (x ) = x² is an even function and g(x) = x is odd then the sum x² + x is neither odd nor even

4 . if f is odd and g is even then the sum f + g is neither odd nor even

b . 1. f ( g ( - x ) ) = f ( g (x ) ) = fg (x ) { if g is even f can be odd or even}

2 . f ( g - x ) ) = f ( - g (x ) ) = fg (x) if g is odd and f iseven

= - fg (x ) if g is odd and f is odd