Answer: x + 3/x -4
Explanation:
(x^2 - 4 x + 3)/x
Roots
x = 1
x = 3
Properties as a real function
Domain
{x element R : x!=0}
Range
{y element R : y<=-2 (2 + sqrt(3)) or y + 4>=2 sqrt(3)}
Derivative
d/dx(x - 3/x - 1 + 6/x - 3) = 1 - 3/x^2
Indefinite integral
integral(-4 + 3/x + x) dx = 1/2 (x - 8) x + 3 log(x) + constant
(assuming a complex-valued logarithm)
Local maximum
max{x - 3/x - 1 + 6/x - 3} = -2 (2 + sqrt(3)) at x = -sqrt(3)
Local minimum
min{x - 3/x - 1 + 6/x - 3} = 2 (sqrt(3) - 2) at x = sqrt(3)
Series representations
-4 + 3/x + x = sum_(n=-∞)^∞ ( piecewise | -4 | n = 0
1 | n = 1
3 | n = -1) x^n
-4 + 3/x + x = sum_(n=-∞)^∞ ( piecewise | 3 (-1)^n | n>1
0 | n = 0
-2 | n = 1) (-1 + x)^n
Definite integral
integral_1^3 (-4 + 3/x + x) dx = log(27) - 4≈-0.704163