Domain and Range

Question

I need help finding the domain and range of this function

 

y=(-4/x-4)+4

Answer 1

y = -4/(x-4) + 4

 

Let's make this one single fraction

y = -4/(x-4) + (4(x-4))/(x-4) = (-4 + 4x -16)/(x-4) = (4x-20)/(x-4)

 

For the domain we need to determine what values of x are not allowed.

Since we cannot divide by 0 we see that x≠4

There is a vertical asymptote at x=4  so the domain is all

real numbers except 4 ...   D = (-∞,4)∪(4,∞) in interval notation

 

For the range we need to determine what restrictions there are on the

output values for y.  We check for horizontal asymptotes (values which y can approach,

but not actually be equal to)

 

Look at the fraction (4x-20)/(x-4)

For horizontal asymptotes the basic rules are

1) If the order of the numerator (highest exponent in the numerator) is greater than

    the order of the denominator then there is no horizontal asymptote

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

    divide the coefficient of the highest exponent variables in the numerator and denominator

    and this will be the asymptote

3)  If the order of the denominator is greater than the order of the numerator

     then the horizontal asymptote will be at y=0

 

We see that the order of the numerator and denominator are both 1 because the

highest exponent of both is x¹ ... So divide the coefficients of the x terms giving

4/1 = 4

There is a horizontal asymptote at y=4 so y can be all real numbers except 4

R = (-∞,4)∪(4,∞) 

 

Note that the open parentheses for interval notation indicate that the endpoint is not included