Find the domain and range of

f(x)= (2x+6) / ((x^2)-x-12)

Step-by-step explanation:

If you want to determine the domain and range of this analytically, you first need to factor the numerator and denominator to see if there is a common factor that can be reduced away.  If there is, this affects the domain.  The domain are the values in the denominator that the function covers as far as the x-values go.  If we factor both the numerator and denominator, we get this:

Since there is a common factor in the numerator and the denominator, (x + 3), we can reduce those away.  That type of discontinuity is called a removeable discontinuity and creates a hole in the graph at that value of x.  The other factor, (x - 4), does not cancel out.  This is called a vertical asymptote and affects the domain of the function.  Since the denominator of a rational function (or any fraction, for that matter!) can't EVER equal 0, we see that the denominator of this function goes to 0 where x = 4.  That means that the function has to split at that x-value.  It comes in from the left, from negative infinity and goes down to negative infinity at x = 4.  Then the graph picks up again to the right of x = 4 and comes from positive infinity and goes to positive infinity.  The domain is:

(-∞, 4) U (4, ∞)

The range is (-∞, ∞)

If you're having trouble following the wording, refer to the graph of the function on your calculator and it should become apparent.

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