Combining Functions, Biocalculus help?

Question

Given that a(x) is a function that is neither even nor odd, and b(x) is an even function, is a ° b(x) even, odd, neither, or is it impossible to tell?

I know that:
Multiplying even functions results in an even function.
Multiplying odd functions results in an even function.
Multiplying an even and odd function results in an odd function.

I'm having a hard time determining whether the resulting function should be even, odd, or neither. I'm leaning towards neither since I can't find any information in my textbook or notes on multiplying an even/odd and neither function. What are the "rules" when it comes to neither function? Can you explain to my why your answer would be the case; or what would be the justification for your answer?

Answer 1

　

Since a is neither even nor odd, then

a(−x) ≠ a(x)

a(−x) ≠ −a(x)



Since b is an even function, then

b(−x) = b(x)



Now (a°b)(x) is NOT the product of 2 functions.

It is the composition of 2 functions: (a°b)(x) = a(b(x))



So let's see what happens when we replace x with −x

(a°b)(−x) = a(b(−x)) ----> since b is even, b(−x) = b(x)

(a°b)(−x) = a(b(x)) = (a°b)(x)

(a°b)(−x) = (a°b)(x)



The function composition (a°b)(x) is even.

Answer 2

math is for *******

Answer 3

neither