Question

Given that f(x) is even and g(x) is odd, determine whether each function is even, odd, or neither. (f • g)(x) = ? (g • g)(x) = ?

Answer 1

Answer:

(f · g)(x) is odd

(g · g)(x) is even

Step-by-step explanation:

If f(x) is even, then f(-x) = f(x).

If g(x) is odd, then g(-x) = -g(x).

(f · g)(x) = f(x) · g(x)

Check:

(f · g)(-x) = f(-x) · g(-x) = f(x) · [-g(x)] = -[f(x) · g(x)] = -(f · g)(x)

(f · g)(-x) = -(f · g)(x) - odd

(g · g)(x) = g(x) · g(x)

Check:

(g · g)(-x) = g(-x) · g(-x) = [-g(x)] · [-g(x)] = g(x) · g(x) = (g · g)(x)

(g · g)(-x) = (g · g)(x) - even

Answer 2

Answer:

(f*g)(x) is odd

(g*g)(x) is even

Step-by-step explanation:

A function is even when f(-x) = f(x) and is odd when g(-x) = -g(x).

(f*g)(-x) = f(x)*[-g(x)] = -[f(x)*g(x)]

That mean (f*g)(x) is odd. For example, take f(x) = x^2 and g(x) = x^3, f(x)*g(x) = x^5, which is odd.

(g*g)(-x) = g(-x)*g(-x) = [-g(x)]*[-g(x)] = g(x)*g(x)

That mean (g*g)(x) is even. For example, take g(x) = x^3, g(x)*g(x) = x^6, which is even.