Determine which functions are ODD (symmetry with origin):

a. f(x) = 2x^3 + x + 3

b. f(x) = 2x^3 + X

c. f(x) = 2x^3 + x^2 + x + 3

d. f(x) = 4x

e. f(x) = x^3 + the absolute value of x

f. f(x) = 2x + 1/x

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A real valued function is said to be odd when the following condition satisfies.

`-f(x) = f(-x)`

Let us consider our functions.

`a. f(x) = 2x^3 + x + 3`

`f(-x) = -2x^3-x+3 != -f(x) `

`b. f(x) = 2x^3 + x`

`f(-x) = -2x^3-x = -(2x^3+x) = -f(x)`

`c. f(x) = 2x^3 + x^2 + x + 3`

`f(-x) = -2x^3+x^2-x+3 != -f(x)`

`d. f(x) = 4x`

`f(-x) = -4x = -f(x)`

`e. f(x) = x^3 +|x|`

`f(-x) = -x^3+|-x| = -x^3+|x| != -f(x)`

`f. f(x) = 2x + 1/x`

`f(-x) = -2x-1/x = -(2x+1/x) = -f(x)`

** So the odd functions are the functions at b,d and f**.

*Note:*

*If you have a constant term in your function that will never be a odd function (as a and c)*

*If you have a even degree polynomial you will never get a odd function.*

**Sources:**

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