Find the domain and range of the function

F(X) = x+4
------
X^2−1

the domain is all reals except where the denominator is zero.

see more information at

http://www.wolframalpha.com/input/?i=%28x%2B4%29%2F%28x^2-1%29

To find the domain and range of a function, we first need to understand the restrictions on the input values (domain) and the output values (range) of the function.

For this function, F(x) = (x+4)/(x^2−1), we need to consider two things:
1. The denominator of the fraction cannot be zero since division by zero is undefined.
2. We need to find the possible output values of the function (range).

Let's start with finding the domain:

Step 1: Denominator cannot be zero
The denominator of F(x) is x^2−1. We know that division by zero is undefined, so we need to exclude values of x for which the denominator is zero. In this case, the denominator is zero when x^2−1=0.

Solving x^2−1=0:
x^2 = 1
Taking the square root of both sides:
x = ±1

So, the values of x that make the denominator zero are x = -1 and x = 1. We need to exclude these values from the domain.

Step 2: Finding the domain
The domain is the set of all real numbers except for the values that make the denominator zero (-1 and 1, in this case). So, the domain of F(x) is all real numbers except x = -1 and x = 1.

Domain: (-∞, -1) U (-1, 1) U (1, ∞)

Now, let's move on to finding the range:

To find the range, we need to determine the possible output values of the function.

Step 1: Analyzing the behavior of the function
Since F(x) is a rational function, we should examine the behavior as x approaches positive and negative infinity.

As x approaches infinity, both the numerator (x+4) and denominator (x^2−1) increase without bound. Therefore, F(x) approaches 0 (the fraction approaches 0).

As x approaches negative infinity, both the numerator (x+4) and denominator (x^2−1) decrease without bound. Again, F(x) approaches 0.

Based on this analysis, the range of F(x) should include 0.

Step 2: Determine if there are any additional restrictions
To check if there are any other restrictions on the range, we can analyze the graph of the function or consider any possible vertical asymptotes.

Step 3: Finding the range
Taking into account the behavior of the function and the fact that there are no vertical asymptotes or other restrictions, we can conclude that the range of F(x) is all real numbers excluding 0.

Range: (-∞, 0) U (0, ∞)

In summary:
Domain: (-∞, -1) U (-1, 1) U (1, ∞)
Range: (-∞, 0) U (0, ∞)