List the intercepts and test for symmetry y= (-5)x^7 over x^8-8

Question

x-intercept, y-intercept and symmetry

Answer 1

f(x)=-5x⁷/(x⁸-8).

When x=0, f(0) is the y intercept=0.

When f(x)=0, x=0, so both intercepts are zero.

There are asymptotes at x=±⁸√8. Also, when x is very large and positive, f(x)≃-5/x→0⁻ and when x is very large and negative, f(x)→0⁺.

f(x) is an odd function:

If x=c², f(c²)=-5c¹⁴/(c¹⁶-8), and f(-c²)=5c¹⁴/(c¹⁶-8). Therefore f(x)=-f(-x), which is asymmetrical. Symmetry is produced by double reflection in the axes (double flip): reflection in the y-axis then in the x-axis, or vice versa.