Question

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

Answer 1

Vertical stretch and reflection across the x-axis are the odd functions.

What are odd functions

Odd functions exhibit symmetry about the origin (0,0) on a graph. They satisfy the condition
`\( f(-x) = -f(x) \)` for all x in their domain. This symmetry means that for any point (x, y) on the function's graph, the point (-x, -y) also belongs to the graph.

Examples of odd functions include
`\( f(x) = x^3` and
`\( f(x) = \sin(x) \)`. Their behavior around the origin shows that as x approaches 0 from the positive and negative sides, f(x) approaches 0, albeit from different sides of the x-axis.

Answer 2

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

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.