How do you determine if #f(x)=-sinx# is an even or odd function?

Question

Cora Alexander · Accepted Answer

f(x) is an odd function.

To determine if f(x) is even/odd consider the following. • If f(x) = f( -x) then f(x) is even ,#AAx" in the domain"# Even functions are symmetrical about the y-axis. • If f( -x) = - f(x) then f(x) is odd, #AAx" in the domain"# Odd functions have half-turn symmetry about the origin. Test for even #f(-x)=-sin(-x)=-(-sinx)=sinx# Since f(x) ≠ f( -x) then f(x) is not even. Test for odd #-f(x)=-(-sinx)=sinx# Since f( -x) = - f(x) then f(x) is odd. graph{-sinx [-10, 10, -5, 5]}

Anna Allen · Answer

undefined To determine if $ f(x) = -\sin(x) $ is an even or odd function, we can apply the following tests:

1. Even Function Test:
   - An even function satisfies the condition: $ f(x) = f(-x) $.
   - For $ f(x) = -\sin(x) $, we have $ f(-x) = -\sin(-x) = -(-\sin(x)) = \sin(x) $.
   - Since $ f(x) $ is not equal to $ f(-x) $ (as $ f(-x) = \sin(x) $), $ f(x) $ does not pass the even function test.

2. Odd Function Test:
   - An odd function satisfies the condition: $ f(x) = -f(-x) $.
   - For $ f(x) = -\sin(x) $, we have $ -f(-x) = -(-\sin(-x)) = \sin(-x) = -\sin(x) $.
   - Since $ f(x) $ is equal to $ -f(-x) $, $ f(x) $ passes the odd function test.

Therefore, $ f(x) = -\sin(x) $ is an odd function.