To determine which of the following functions is an odd function, we first need to understand what an odd function is. An odd function is one that satisfies the condition: f(-x) = -f(x) for all x in its domain. This means that the graph of the function is symmetric with respect to the origin.
Now, let’s consider a few examples of functions:
- f(x) = x^3
- g(x) = x^2
- h(x) = sin(x)
- k(x) = cos(x)
Let’s analyze them:
- For f(x) = x^3:
f(-x) = (-x)^3 = -x^3 = -f(x). This is an odd function. - For g(x) = x^2:
g(-x) = (-x)^2 = x^2 = f(x). This is not an odd function. - For h(x) = sin(x):
h(-x) = sin(-x) = -sin(x) = -h(x). This is also an odd function. - For k(x) = cos(x):
k(-x) = cos(-x) = cos(x) = f(x). This is not an odd function.
From the above analysis, we can conclude that f(x) = x^3 and h(x) = sin(x) are odd functions. Thus, if you need to select which of the functions listed is an odd function, you would choose x^3 or sin(x).