Identify the asymptotes, domain, and range of the function. f(x)= 1/ (x + 7) - 8

Question

Can Someone Solve this for me please and show steps i need this to study for finals.

Answer 1

y=1/(x+7)-8

x+7=0, x=-7 is the vertical asymptote

the function can be written as y=(1-8x-56)/(x+7)=(-8x-55)/(x+7) (find a common denominator and add the 2 terms)

because the degrees of the numerator and denominator are equal, the horizontal asymptote is y=the fraction of the coefficients of the x terms, y=-8/1 or y=-8

the domain is all real numbers except what will make the denominator equal to 0

so, the domain is all real numbers except -7

to find the range, find the inverse of the function and its domain will be the range of the original function

interchange the x and y in the original function and solve for y to get the inverse

the domain of the inverse is the range of the original

after doing all of this you get...

y=(-55-7x)/(x+8)

the domain is all real numbers except -8 so the range of the original function is all real numbers except -8

another way to look at this is in the function y=(1/(x+7)-8, y can never be -8 because if y were -8 then

1/(x+7) would have to be equal to 0 but 1/(x+7) can never be equal to 0, so y can be any number except -8