Question

Does the graph of y = sin x + cos x have even symmetry, odd symmetry, or neither?

Answer 1

\[f(x)=\sin(x)+\cos(x) \\ \text{ plug \in } -x \\ f(-x)=? \\ \text{ if you get } f(-x)=f(x) \text{ then } f \text{ is even } \\ \text{ if you get } f(-x)=-f(x) \text{ then } f \text{ is odd }\]

Answer 2

f(-x)=sin(-x)+cos(-x)
now what? @freckles

Answer 3

use the fact that sin is an odd function
and that cos is an even function

Answer 4

that is you are using that sin(-x)=-sin(x)
and that cos(-x)=cos(x)

Answer 5

okay i got
cos(x)=f(-x)+sin(x)
@freckles

Answer 6

did you add sin(x) on both sides?

Answer 7

\[f(-x)=\sin(-x)+\cos(-x) \\ f(-x)=-\sin(x)+\cos(x) \]
is -sin(x)+cos(x) the original function or the opposite of the original function?

Answer 8

or neither?

Answer 9

@freckles -sin(x)+cos(x) is the opposite of the original function

Answer 10

\[-\sin(x)+\cos(x) \neq \sin(x)+\cos(x) \text{ the signs on } \sin \text{ differ } \\ -\sin(x)+\cos(x) \neq -(\sin(x)+\cos(x)) \text{ both the signs would have to be differ}\]
what I'm saying is that
the opposite of sin(x)+cos(x) is -(sin(x)+cos(x)) but we did not get this so it is not odd
and also we did not get sin(x)+cos(x) back so we know this is not even

Answer 11

another way to say -(sin(x)+cos(x))
is -sin(x)-cos(x)
this is what I meant by both of the signs needed to be different

Answer 12

but anyways
in summary
-sin(x)+cos(x) is neither the same as or the opposite of sin(x)+cos(x)
so sin(x)+cos(x) is neither even or odd

Answer 13

if we had:


\[f(x)=x^2+\cos(x) \\ \text{ and we plug in } -x \\ f(-x)=(-x)^2+\cos(-x) \\ f(-x)=x^2+\cos(x) \\ \text{ this would mean } f \text{ is even } \\ \text{ if we had } f(x)=x^3+\sin(x) \\ \text{ and we plug in } -x \\ \text{ we would have } \\ f(-x)=(-x)^3+\sin(-x) \\ f(-x)=-x^3-\sin(x) \\ f(-x)=-(x^3+\sin(x)) \text{ this is the opposite of } f \\ \text{ so this } f \text{ would be odd }\]

Answer 14

anyways was just giving examples of an odd and even function just in case you were confused or whatever

Answer 15

let me know if you still don't understand

Answer 16

no i think i understand now. thank you so much @freckles

Answer 17

np