The domain of a function is all possible values of x xwhere f(x)f(x) is defined. Here, when the denominator is equal to 0, the function is undefined. In this case:
x+6=0x+6=0
x=-6x=−6
So yy is only undefined at x=-6x=−6. In interval notation, we write the domain as (-oo,-6)uu(-6,oo)(−∞,−6)∪(−6,∞).
The range of a function is all possible values for yy. Another way to solve for the range is to find y^-1y−1, as in, the inverse function of yy, and find its domain. Here, we can find y^-1y−1:
y=3/(x+6)y=3x+6
Switch the variables and solve for yy:
x=3/(y+6)x=3y+6
1/x=(y+6)/31x=y+63
3/x=y+63x=y+6
y=3/x-6y=3x−6
This is y^-1y−1. It is also defined when its denominator is equal to 00.
So y^-1y−1 will be undefined when x=0x=0, and therefore yy will be undefined when y=0y=0. The range in interval notation is:
(-oo,0)uu(0,oo)(−∞,0)∪(0,∞)
