The range of a function refers to the set of all possible output values that the function can produce for its corresponding input values. In this case, the function is:
f(x) = 6x / (x^2 - 9)
To find the range of this function, we need to consider the restrictions on the denominator (x^2 - 9) that would make the function undefined.
The denominator (x^2 - 9) can be factored as a difference of squares:
x^2 - 9 = (x + 3)(x - 3)
So, the function is undefined when the denominator is equal to zero, i.e., when:
(x + 3)(x - 3) = 0
This occurs when x = -3 or x = 3.
Now, we need to consider the behavior of the function as x approaches these values. As x approaches -3 or 3 from the left, the function becomes increasingly negative, and as x approaches -3 or 3 from the right, the function becomes increasingly positive. This is because the numerator (6x) remains constant while the denominator approaches zero, resulting in a very large positive or negative value for f(x).
Since the function can approach arbitrarily large positive or negative values as x approaches -3 or 3, respectively, the range of the function is (-∞, ∞), which represents all real numbers except for the values -3 and 3. In interval notation, the range of the function can be written as:
Range of f(x): (-∞, ∞) \ {-3, 3}