Question 1167698
arnold's equation is:


f(x) = x - 2/x


let y = f(x).


the equation becomes y = x - 2/x


place all terms under a common denominator to get:


y = (x^2 - 2)/x


the vertical asymptote is at x = 0.


since the degree of the numerator is 1 more than the degree of the denominator, the asymptote will be found by dividing the numerator by the denominator and using the quotient in the equation of the asymptote.


y = (x^2 - x) / x = x with a remainder of -2.


the quotient is x and the remainder is -2.


the equation of the quotient is y = x


the domain of the function is all values of x except at x = 0.


the range of the function is all values of y.


as x approaches 0 from the left, y approaches minus infinity.


as x approaches 0 from the right, y approaches plus infinity.


the slant asymptote is the line y = x.


this is what the graph looks like.


<img src = "http://theo.x10hosting.com/2020/102002.jpg" >


beth's equation is:


g(x) = 3 - 2/x


replace g(x) with y to get:


y = 3 - 2/x


place all terms under a common denominator to get:


y = (3x - 2) / x


the degree of the numerator is the same degree as the denominator, so there will be a horizontal asymptote.


divide 3x - 2 by x to get a quotient of 3 with a remainder of 2.


the equation of the asymptote will be y = 3


this is what the graph looks like.


<img src = "http://theo.x10hosting.com/2020/102003.jpg" >


both equations have a vertical asymptote at x = 0


therefore, the domain of both equations is all values of x except at x = 0.


the range of both functions is equal to all real values of y.


the fact that y is undefined at x = 0 doesn't affect the range of the equations.


there are no values of y that are not in the range of y .


in addition to the vertical asymptotes, .....


arnold's equation has a slant asymptote at y = x.


beth's equation has a horizontal asymptote at y = 3


here's a reference on horizontal and slant asymptotes.


<a href = "https://www.purplemath.com/modules/asymnote.htm" target = "_blank">https://www.purplemath.com/modules/asymnote.htm</a>