How do you find the domain and range of f(x)= 2- 1/(x+6)^2?

1 Answer
May 1, 2018

Domain: (-infty, -6) cup (-6, infty)
Range: (-infty, 2)

Explanation:

The domain is the set of all possible inputs x to a function that give a valid result. Ie, all of the numbers you can replace x with and not break any algebraic rules is the domain.

Here, we note that we have the term 1/(x+6)^2 in our function. We know that we cannot divide by zero, so it follows that (x+6)^2 ne 0. This implies that x ne -6. Thus, x = -6 is not included in our domain. All other values of x provide a legal answer, however, so our domain is (-infty, -6)cup(-6, infty).

The range is the set of all possible outputs of a function. That is, if you plug in every x from the domain into f(x), all the results you get consist of the range.

We note that when x gets very large, the denominator of 1/(x+6)^2 gets very large, making the term itself quite small. Thus, f(x) will get very near 2 with large values of x.

With values of x very close to -6, we find that the denominator of 1/(x+6)^2 gets very small, making the term evaluate to a large number. For closer and closer values of x to -6, f(x) = 2 - 1/(x+6)^2 begins to get very negative, tending toward negative infinity.

Thus, our function covers all values between -infty and 2, though it never quite reaches 2. (And it cannot reach -infty.) So our range is (infty, 2).

A graph of this function reveals this behavior.

graph{2 - 1/(x+6)^2 [-16, 16, -8, 8]}