describe the symetry of y=x^3-x^5+x
To understand the symmetry of the graph of the function y = x^3 - x^5 + x, we need to analyze its properties based on evenness and oddness.
1. Evenness: A function is even if it remains unchanged when you replace "x" with "-x". To check for evenness, substitute "-x" in place of "x" in the function and simplify:
y = (-x)^3 - (-x)^5 + (-x)
= -x^3 - (-x)^5 - x
= -x^3 + x^5 - x
By comparing the original function (y = x^3 - x^5 + x) with the simplified form, we can see that the signs of all terms have changed. Therefore, the function is not even.
2. Oddness: A function is odd if it remains unchanged when you replace "x" with "-x" and multiply the whole expression by -1. To check for oddness, substitute "-x" in place of "x" and multiply the expression by -1:
y = -(-x^3 - (-x)^5 - x)
= x^3 - x^5 + x
By comparing the original function (y = x^3 - x^5 + x) with the simplified form, we can conclude that they are identical. Therefore, the function is odd.
Based on this analysis, we can say that the function y = x^3 - x^5 + x is odd, meaning it exhibits symmetry with respect to the origin (0,0). When you reflect the graph of the function across the origin, it will produce an identical graph.