Give an odd function which of following transformations would result in an odd function

Reflection across the x-axis and Vertical stretch are the ones that return an odd function.

Step-by-step explanation:

An odd function f(x) is a function such that:

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

So let's analyze the options:

A) Horizontal translation.

An horizontal translation of A units to the right (A > 0) is written as:

g(x) = f(x  - A)

Now, let's see if g(x) is also odd.

g(x) = f(x - A)

g(-x) = f(-x - A)

Now, f(-x - A) is equal to -f( x + A)

then:

g(-x) = f(-x - A) ≠ -g(x) = -f(x - A)

This is not an odd function.

B) Reflection over the x-axis.

When we have a point (x, y), a reflection over the x-axis changes the sign of the y-variable.

Then we have that:

g(x) = -f(x).

Then:

g(x) = -f(x)

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

then:

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

This is an odd function.

C) Vertical stretch:

We can write a vertical stretch of factor scale A as:

g(x) = A*f(x).

Let's see if g(x) is odd:

g(-x) = A*f(-x) = A*(-f(x)) = -A*f(x) = -g(x)

this is a odd function.

C) Vertical translation:

A vertical translation of A units up (A > 0) is written as:

g(x) = f(x) + A.

Similar to the case of the horizontal translation, so it is easy to see that g(x) is not an odd function.