Answer: 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.
