Question

Which of the following is an odd function?

Answer 1

Option 4

f(x) = -x is an odd function

Solution:

A function is odd if and only if f(–x) = –f(x)

Option 1

$f(x) = x^3 + 5x^2 + x$

Substitute x = -x in above equation

$f(-x) = (-x)^3 + 5(-x)^2 + (-x)$

Cubes always involve multiplying a number by itself three times, so if the number is negative the cube will always be negative

Ans squaring results in positive

$f(-x) = -x^3 + 5x -x$  --- eqn 1

$-f(x) = -(x^3 + 5x^2 + x)\\\\-f(x) = -x^3 - 5x^2 - x$  ---- eqn 2

Comparing eqn 1 and eqn 2,

$f(-x) \neq -f(x)$

Therefore not an odd function

Option 2

$f(x) = \sqrt{x}$

$f(-x) = \sqrt{-x}$

$-f(x) = - \sqrt{x}$

Therefore,

$f(-x) \neq -f(x)$

Therefore not an odd function

Option 3

$f(x) = x^2 + x\\\\f(-x) = (-x)^2 + (-x)\\\\f(-x) = x^2 - x$

$-f(x) = -(x^2 + x) = -x^2 - x$

$f(-x) \neq -f(x)$

Therefore not an odd function

Option 4

$f(x) = -x\\\\f(-x) = -(-x) = x$

$-f(x) = -(-x) = x$

$f(-x) = -f(x)$

Thus option 4 is correct and it is an odd function