The graph of the equation y=x^3-x is symmetric with respect to which of the following?

a) the x-axis
b) the y-axis
c) the origin
d) none of these

Answer: c

2) The graph of an odd function is symmetric with respect to which of the following?
a) the x-axis
b) the y-axis
c) the line y=x
d) none of these

Thanks

In both case I agree. In 1) the function is odd and therefore symmetric about the origin

In 2)again an odd function is symmetric about the origin, so none of the above

Definition of Odd Function

A function with a graph that is symmetric with respect to the origin. A function is odd if and only if f(–x) = –f(x).

Oops, forgot to mention that the answer for problem 2 is d.

You're welcome! And for the second question, the answer is a) the x-axis. The graph of an odd function is like that one friend who always keeps you grounded - unconditionally symmetric with respect to the x-axis.

To determine the symmetry of a graph with respect to a specific line or axis, you have to consider the equation or the characteristics of the function.

1) For the equation y = x^3 - x, we can analyze it to determine its symmetry. Since the function contains both an odd (x^3) and an even (-x) term, we need to evaluate its symmetry separately for the x-axis and the y-axis.

To determine symmetry with respect to the x-axis, we substitute -x for x in the equation and check if it remains the same:
y = (-x)^3 - (-x)
y = -x^3 + x
This equation is not the same as the original equation (y = x^3 - x). Therefore, the graph is not symmetric with respect to the x-axis (option a).

To determine symmetry with respect to the y-axis, we replace x with -x in the equation and check if it remains the same:
y = (-x)^3 - (-x)
y = -x^3 + x
By comparing this equation to the original equation, we can see that they are equivalent. Therefore, the graph is symmetric with respect to the y-axis (option b) since all points (x, y) on the graph have corresponding points (-x, y) on the opposite side.

To determine symmetry with respect to the origin, we substitute -x for x and -y for y in the given equation:
-y = (-x)^3 - (-x)
-y = -x^3 + x
Rearranging the equation gives us:
y = x^3 - x
This equation is the same as the original equation. Therefore, the graph is symmetric with respect to the origin (option c) since all points (x, y) have corresponding points (-x, -y) on the opposite side.

Hence, the correct answer is option c: the origin.

2) For an odd function, its graph is symmetric with respect to the origin. This is because odd functions satisfy the property f(-x) = -f(x), which means that for any point (x, y) on the graph, the point (-x, -y) will also be on the graph. This symmetry occurs only if we reflect the graph about the origin.

Therefore, the correct answer is option d: none of these, as the graph of an odd function is not symmetric with respect to the x-axis, y-axis, or the line y = x.

I hope this explanation helps! Let me know if you have any further questions.