Question

the function y= x/tanx is ( an even, an odd, neither an even or odd) function, the function y= secx/x is (an even, an odd, neither an even or odd) function? I have to select one of the answers in parentheses.

Answer 1

y = x/tanx is neither odd nor even

y=secx/x is an odd function

Explanation:

these are the answers for PLATO

Answer 2

If
`v(x)=(x)/(\tan(x))`, then
`v` is even.

If
`w(x)=(\sec(x))/(x)`, then
`w` is odd.

Explanation:

Summary of rules/ what we need:

`f(-x)=f(x)` implies
`f` is even.

`f(-x)=-f(x)` implies
`f` is odd.

So in either case, we need to replace
`x` with
`-x`.

Let's begin.

First Problem:

`v(x)=(x)/(\tan(x))`

Replace
`x` with
`-x`:

`v(-x)=(-x)/(\tan(-x))`

`v(-x)=(-x)/(-\tan(x))` (We used
`\tan(x)` is odd; that is,
`\tan(-x)=-\tan(x)`)

`v(-x)=(x)/(\tan(x))`

`v(-x)=v(x)`

This implies
`v` is an even function.

Second Problem:

`w(x)=(\sec(x))/(x)`

Replace
`x` with
`-x`:

`w(-x)=(\sec(-x))/(-x)`

`w(-x)=(\sec(x))/(-x)` (We used
`\sec(x)` is even; that is,
`\sec(-x)=\sec(x)`)

`w(-x)=-(\sec(x))/(x)`

`w(-x)=-w(x)`

This implies
`w` is an odd function.