Question

Which of the following is an odd function?

Answer 1

Answer with Step-by-step explanation:

A function f is odd if:

f(-x)= -f(x) for all x in the domain of f

1. f(x)=
`x^3+5x^2+x`

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

=
`-x^3+5x^2-x`

-f(x)=
`-x^3-5x^2-x`

f(-x) ≠ -f(x)

So, function is not odd

2. f(x)=√x

f(-x)=√(-x)

= i√x

-f(x)= -√x

f(-x) ≠ -f(x)

So, function is not odd

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

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

=
`x^2-x`

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

f(-x) ≠ -f(x)

So, function is not odd

4. f(x)= -x

f(-x) = -(-x)

=x

-f(x)=x

f(-x) = -f(x)

So, function is odd

Answer 2

f(x) = -x is odd

Explanation:

A function f is odd if f(-x) = -f(x) for all x in the domain of f. Then, let's check each of the functions.

1. f(x) = x^3 + 5x^2 + x

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

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

Given that f(-x) ≠ -f(x). The function f is not odd.

2. f(x) = sqrt(x)

f(-x) = sqrt(-x) (Imaginary number)

-f(x) = -sqrt(x)

Given that f(-x) ≠ -f(x). The function f is not odd.

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

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

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

Given that f(-x) ≠ -f(x). The function f is not odd.

4. f(x) = -x

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

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

Given that f(-x) = -f(x). The function f is odd.