I need help finding the domain and range of these functions

Question

y=(1/x)-2 and also y=(-4/x-4)+4

 

also the problem told me to compare 

Answer 1

The domain is the values of x where the graph exists, and the range is the values of y where the graphs exists.

 

y = (1/x) - 2

 

We have a rational term in this function (1/x).  We know that when the denominator of a fraction is zero, the number is undefined. To find the domain, set the denominator equal to to zero.

 

x = 0

 

So the domain in this function is all real numbers except for 0.

 

x = 0 is also a vertical asymptote.

 

 

To find the range of the function, find the horizontal asymptotes. We do this by examining the degrees of the leading terms.  If the degree of the numerator is less than the degree of the denominator in the function, then the horizontal asymptote is 0.  If the degree of the numerator is greater than the degree of the denominator in the function, then there is no horizontal asymptote.  If the degree of the numerator is equal to the degree of the denominator in the function, then divide their coefficients.  

 

First, lets rewrite the function in rational form.

 

y = (1 - 2x) / x

 

y = (-2x + 1) / x

 

We look at -2x in the numerator and x in the denominator.  The degrees are the same because the exponent of x is 1, so we just divide their coefficients.

 

-2 / 1  = -2

 

The horizontal asymptote is  y = -2

 

The range is all real numbers except for -2.

 

 

Now try out the next problem on your own.