graph rational function

Question

The following rational function 

f(x)=(2x-1)/(x-1)

has horizontal asymptote y=

has vertical asymptote x=

y intercept is y=

x intercept is x=

 

Answer 1

The function f(x) has a vertical asymptote when the denominator is zero, at x = 1.  

 

The function has an horizontal asymptote also.  To find it, do the division and get (2x-1)/(x-1) = 2 + 1/(x-1). Since 1/(x-1) approaches zero as x goes to infinity or -infinity, the function has an assymptote to the horizontal line f(x) = y = 2.  

 

So the graph has two parts, one on each of the two parts of the domain, (-infinity,1)  and (1,infinity).  To the left of x=1 the graph goes to -infinity as x approaches 1 and crosses the x axis at x = 1/2.    On the right of x = 1 the graph approaches +infinity as x approaches 1 and it's value is always greater than 2.  

 

The left-hand part of the graphs has x-intercept is x = 1/2 and y-intercept y = 1.  The right-hand part has no x-intercept or y-intercept.

 

 

Answer 2

Hi Dalia;

The horizontal asymptote can be easily established by the fact that the x in the numerator and the x in the denominator are of the same degree, 1.  Henceforth, we need only divide the coefficients of x...

y=2/1=2.....horizontal asymptote

The vertical asymptote is a result of the denominator being equal to zero, undefined.

x-1=0

x=1....vertical asymptote

 

f(x)=(2x-1)/(x-1)

y=(2x-1)/(x-1)

y-intercept is the value of y when x=0...

y=[(2)(0)-1]/(0-1)

y=-1/-1

y=1

y-intercept, (0,1)

 

 

y=(2x-1)/(x-1)

x intercept is the value of x when y=0...

0=(2x-1)/(x-1)

0=2x-1

1=2x

1/2=x

x-intercept, (1/2,0)

 

 

 

 

Answer 3

The easy way to find the horizontal asymptote is to consider what happens to the function when x becomes very large (say 1000).   For large x, the -1 in the numerator can be neglected, as can the -1 in the denominator.  Thus for large x, the function becomes 2 x/ x = 2.   So the horizontal asymptote is y = 2

 

A vertical asymptote comes about when the denominator becomes zero.   This happens at x = 1, so the vertical asymptote is x = 1.

 

The y intercept is f(0)  =  -1/-1 = 1.      (or y = 1)

 

The x intercept is the solution of f(x) = 0 .  As long as the denominator is not zero, we can just look for the x value for which the numerator is zero.    This x value is x = 1/2.    So  x intercept:  x = 1/2

 

In the more general case there can be more than one vertical asymptote and even two horizontal asymptotes (one for x becoming large positive, the other for x becoming large negative).   An example of this is the arctan function, which has horizontal asymptotes at - pi/2 and + pi/2