Question

let f(x) be an even function and g(x0 be an odd function. use the definition of even and odd functions to determine whether the functions are even odd or neither .

1. F(x)*G(x)

2. F(x)/G(X)

3. F(G(x))

4. G(F(x))

5. F(x)+G(x)

6. F(x)-G(x)

Answer 1

f(x) is an even function ⟹ f(-x) = f(x)

g(x) is an odd function ⟹ g(-x) = -g(x)

Answer 2

1. f(x)*g(x)

to find if this product of functions is odd or even
compute

f(-x)*g(-x)

and see if is equal or opposite to f(x)*g(x)

Answer 3

@Reggie97g

Answer 4

i'm very confused on how you know if its odd or even . is it like positive or negative ?

Answer 5

Some function A(x) is even, means that A(-x) = A(x)

Some function B(x) is odd, means that B(-x) = -B(x)

Answer 6

It might be more clear if , for the first question

1. f(x)*g(x)

we set h(x) = f(x)*g(x)

and then compute h(-x), in-terms of f(x) and g(x)


if we find that h(-x) = h(x) then we have shown h(x) is even
if we find that h(-x) = -h(x) then we have shown h(x) is odd

Answer 7

h(x) = f(x)*g(x)

h(-x) = f(-x)*g(-x)
= ( use the fact that f(x) is even, and g(x) is odd )
=

Answer 8

can you work out the next line @Reggie97g ?

Answer 9

h(x) = f(x)*-g(x) ?

Answer 10

h(-x) = f(-x)*g(-x)
you mean
h(-x) = f(x) * -g(x) right?

Answer 11

oops yes

Answer 12

so is h(-x) equal or opposite to h(x) ?

Answer 13

it is opposite right ?

Answer 14

no no its equal i just saw what you put sorry

Answer 15

h(-x) = f(x) * -g(x)
= -f(x)*g(x)
= -h(x)

h(-x) is opposite to h(x)

Answer 16

but before you put h(-x)=h(x) !

Answer 17

ooh that shows its even and not odd

Answer 18

is h(x) even or odd?

Answer 19

even

Answer 20

how did you get that?

Answer 21

if h(-x) = -h(x) then its odd and thats what the problem equals . so the function must be odd

Answer 22

thats better

Answer 23

For the next one

h(x) = f(x) / g(x)

h(-x) = f(-x) / g(-x)
= (simplify this )

Answer 24

(remember, you know f(x) is even, and you also know g(x) is odd)

Answer 25

= f(x) / -g(x)
= -h(x) so its odd

Answer 26

great!, so far you have proved that

1. the product of an even function and an odd function is and odd function
2. the quotient of an even function and an odd function is and odd function

Answer 27

3. is the composition of functions

this time
h(x) = f(g(x)) = f ∘ g (x)

the method is the same

find h(-x)

Answer 28

*an