which of these is an odd function ?

Question

hi,

i would like to find out how to solve this problem (i am asking this question again as i got not so many responses before)
given the functions f(x) = cos x + sec x,f(x) = tan x + cos(x^2), f(x) = x+tan( 5 x ) , f(x) = 3 x^3-4 x -9/x and finally f(x) = x^2 + 1 - cos x , which of these would be odd functions.

thanks for your help. (hope that made sense in that above sentence :)

Answer 1

An odd function satisfies the following property:

f(x) = -f(-x) OR f(-x) = -f(x)

An even function satisfies the following property:

f(x) = f(-x)

The odd trigonometric functions are sin(x), tan(x), csc(x), cot(x).
The even trigonometric functions are cos(x), sec(x).

1) f(x) = cos(x) + sec(x)

Calculate -f(-x). This involves substituting a (-x) for every x, and multiplying the result by (-1).

-f(-x) = (-1) (cos(-x) + sec(-x))

But cos(x) and sec(x) are even functions, so
cos(-x) = cos(x), and sec(-x) = sec(x).

-f(-x) = (-1) (cos(x) + sec(x))
= -cos(x) - sec(x), which is NOT equal to f(x). Therefore, this function is not odd.

2) f(x) = tan(x) + cos(x^2)

Test -f(-x).

-f(-x) = - [tan(-x) + cos( [-x]^2 )

tan(x) is odd, so tan(-x) = -tan(x)

-f(-x) = - [-tan(x) + cos (x^2)]
-f(-x) = tan(x) - cos(x^2)
Compare to f(x), and as you can see, they're not equal, so this function is not odd.

3) f(x) = x + tan(5x)

-f(-x) = - [ (-x) + tan(5*(-x)) ]
-f(-x) = - [-x + tan(-5x)]

tan(x) is odd, so tan(-5x) = -tan(5x).

-f(-x) = - [-x - tan(5x)]
-f(-x) = x + tan(5x), which is equal to f(x).

Therefore, f(x) is odd.

4) f(x) = 3x^3 - 4x - 9/x

-f(-x) = - ( 3[-x]^3 - 4[-x] - 9/[-x] )

Note that cubing a negative term doesn't remove the negative sign.

-f(-x) = - ( -3x^3 + 4x + 9/x )
-f(-x) = 3x^3 - 4x - 9/x

This is equal to f(x), so this is odd.

5) f(x) = x^2 + 1 - cos(x)

-f(-x) = - [ (-x)^2 + 1 - cos(-x) ]

cos(x) is even, so cos(-x) = cos(x)

-f(-x) = - [ x^2 + 1 - cos(x)]
-f(-x) = -x^2 - 1 + cos(x).
This is not equal to f(x), so it is not odd.

Answer 2

f(x) = cosx + sec x is an even function since cosx is an even function, 1/cosx would be as well.

f(x)= tan x + cos (x^2) is odd, i believe, since sinx is an odd function and sinx/cos x would be an odd/even thus making it odd.

f(x)= 3x^3-4x-9/x is an odd function since x^3 is an odd function.

f(x)= x^2+1 - cosx is an even function since both x^2 and cosx are even functions.

So, functions 2 and 3 should be the odd ones.

Answer 3

f(x) = cosx + secx is an even function because both its constituent parts are even functions.

f(x) = tanx + cos(x^2) is neither odd nor even because it is the sum of both.

f(x) = x + tan(5x) is an odd function as both its constituent parts are odd functions.

f(x) = 3x^3 - 4x - 9/x is an odd functions because all of its parts are odd functions.

f(x) = x^2 + 1 - cosx is an even function because all of its parts are even functions.

Answer 4

an odd function is a function such that for all x in its domain, f(-x)= -f(x). This is all you need to know to answer any arbitrary question like the ones you gave.
The method is always the same. Simply substitute -x for x in the f(x) and see if it becomes -f(x).

Answer 5

They all look pretty odd to me.