Domain and range of function

Question

What is the domain and range of the function y=(-4/x-4)/4

 

then the problem asks me to compare with a graph of y=1/x

Answer 1

This appears to read as y=(-4/(x-4))/4 = -4/(4(x-4)) = -4/(4x-16)

 

If this is the case then for the domain we need to see what input values for x are allowed

In this case x≠4 since we cannot divide by zero so there is a vertical asymptote at x=4.

This is the only restriction on the domain x is all real numbers except 4,  so D = (-∞,4)∪(4,∞)

The open parentheses for interval notation indicate that the endpoints are not included

 

For the range we need to determine all possible output values y for the function

We need to determine if there are any values that y cannot be...

This requires looking for horizontal asymptotes:

The basic rules for horizontal asymptotes are

 

1)  If the degree of the numerator is greater than that of the denominator... no asymptote

2)  If the degree of the numerator is equal to the degree of the denominator then the

    asymptote will be equal to the coefficient of the highest degree variable in numerator

    divided by the coefficient of the highest degree variable in the denominator

    Example:  (2x²+5)/(5x²-7) has same degree in both numerator and denominator so the

    horizontal asymptote would be at 2/5

3) If the degree of the denominator is greater than the degree of the numerator then the

    horizontal asymptote is at y=0

 

In this case the degree of the numerator is 0, degree of denominator is 1 so the

horizontal asymptote is at y=0

So the graph will approach y=0 without ever actually reaching it

 

Thus the range is all real numbers except y = 0 so   R = (-∞,0)∪(0,∞)

 

 

For the graph of y = 1/x  we have the restriction x≠0 so there is a vertical asymptote at x=0

D = (-∞,0)∪(0,∞)

The range has the same restriction with a horizontal asymptote at y=0 since the degree of

the denominator is greater than the degree of the numerator and

R = (-∞,0)∪(0,∞)