A function f is called odd if f(-x) = -f(x) for every x in its domain, a functio

Question

A function f is called odd if f(-x) = -f(x) for every x in its domain, a function g is called even if g(-x) = g(x) for every x in its domain. Suppose f, g, and h are functions defined for all real numbers so that f and h are odd and g is even. Following the model below, prove the statements a-c. The function f/g is odd. Proof. (f/g)(-x) = f(-x)/g(-x) = -f(x)/g(x) = -(f(x)/g(x)) = -(f/g)(x) A. The function f/h is even. B. The function fg (multiplication of functions) is odd. C. The function f o h (composition of functions) is odd.

Explanation / Answer

given f(x) and h(x) are odd => h(-x) = -h(x) and f(-x) = -f(x) and g(x) is even => g(-x) = g(x) A) f(-x)/h(-x) = -f(x)/-h(x) = f(x)/h(x) => so even B)f(-x)g(-x) = -f(x)*g(x) => so odd C) f( h(-x) ) = f( - h(x) ) = - f(h(x)) => so odd

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