Question

How do you find the domain and range of `y = (x+4 )/( x-4)`?

Answer 1

Domain: `(-oo, 4) uu (4, + oo)`
Range: `(-oo, 1) uu (1, + oo)`

Explanation:

The domain of the function will include all the values of `x` for which the denominator is not equal to zero.

This means that you have

`x - 4 = 0 implies x = 4`

This value of `x` will be excluded from the domain of the function. This implies that the function's domain will be `x in RR "\" {4}`, or `x in (-oo, 4) uu (4, + oo)`.

To find the range of the function, use some algebraic manipulation to rewrite the function as

`y = (x+4)/(x-4) = (x - 4 + 8)/(x-4) = 1 + 8/(x-4)`

Since `8/(x-4) !=0 AA x in (-oo, 4) uu (4, + oo)`, it follows that you can never have `y = 1`.

This means that the range of the function will be `x in RR "\" {1}`, or `x in (-oo, 1) uu (1, + oo)`.

graph{(x+4)/(x-4) [-9.99, 10.02, -5, 4.995]}