determine all symmetries of the function g(x)=x/x-1

well, f(-x) = -x/(-x-1) = x/(x+1) ≠ -f(x)

so no symmetry about (0,0)

Also, f(-x) ≠ f(x), so no symmetry about the y-axis

f^-1(x) = x/(x-1), so there is symmetry about the line y=x.

See
http://www.wolframalpha.com/input/?i=inverse+x%2F%28x-1%29

ok I get it. But what's the actual answer

To determine all the symmetries of the function g(x) = x/(x - 1), we need to analyze its properties and find any transformations that preserve its shape.

1. Reflection Symmetry (y-axis symmetry):
To check for reflection symmetry, replace x with -x in the function and simplify:
g(-x) = (-x)/(-x - 1) = x/(x + 1)
If g(-x) = g(x), the function has reflection symmetry about the y-axis. In this case, g(-x) = g(x) implies x/(x + 1) = x/(x - 1). Cross-multiplying and simplifying, we get: (x - 1)(x + 1) = x(x + 1)
Expanding both sides gives: x^2 - 1 = x^2 + x
Simplifying further, we find: -1 = x
Since there is no value of x that satisfies this equation, the function g(x) = x/(x - 1) does NOT have reflection symmetry about the y-axis.

2. Point Symmetry (origin symmetry):
To check for point symmetry, replace x with -x in the function and simplify:
g(-x) = (-x)/(-x - 1) = x/(x + 1)
If g(-x) = -g(x), the function has point symmetry about the origin (0, 0). In this case, g(-x) = -g(x) implies x/(x + 1) = -x/(x - 1). Cross-multiplying and simplifying, we get: (x - 1)(x + 1) = -x(x + 1)
Expanding both sides gives: x^2 - 1 = -x^2 - x
Simplifying further, we find: 2x^2 + x - 1 = 0
By solving this quadratic equation, we can find the values of x that satisfy it. However, upon solving, we find that these values do not exist. Therefore, the function g(x) = x/(x - 1) does NOT have point symmetry about the origin.

Based on these analyses, we can conclude that the function g(x) = x/(x - 1) does not have any symmetries.