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.

2? if not the correct answer you might need to provide more information

step-by-step explanation:

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