Question

If it is known that function f is odd and function g is even. determine if the following

functions are odd, even, or neither odd nor even.
1/g(x)

Answer 1

The function 1/g(x) is even.

Step-by-step explanation:

Given that function f is odd and function g is even, we want to determine the nature of the function 1/g(x).

An odd function has the property: f(-x) = -f(x), while an even function has the property: g(-x) = g(x).

Using these properties, let's analyze 1/g(x):

1. When we substitute -x in 1/g(x), we get 1/g(-x).
2. Since g(x) is even, g(-x) = g(x), so 1/g(-x) becomes 1/g(x).
3. The fact that 1/g(x) = 1/g(-x) means that the function is even.

Therefore, the function 1/g(x) is even.

Answer 2

`(1)/(g)\ \text{ is an even function}`

Step-by-step explanation:

Considers the function h defined by :

h(x) = 1/g(x)

g is an even function means g(-x) = g(x)

then

h(-x) = 1/g(-x) = 1/g(x) = h(x)

Therefore

h is an even function.