y = (x - 1) / ((4x^2) - 4)
1. Factor the denominator completely. Show your work and explain how you got your answer.
2. List all the discontinuities in this function. Make sure to identify the types of discontinuity (asymptote, removable, jump)
3. What is the domain of this function?
4. What is the range of this function?
To factor the denominator (4x^2 - 4), we can start by factoring out the greatest common factor, which in this case is 4:
4x^2 - 4 = 4(x^2 - 1)
Next, we can notice that (x^2 - 1) is a difference of squares, which can be factored further:
x^2 - 1 = (x - 1)(x + 1)
So the denominator of the function can be factored completely as:
4x^2 - 4 = 4(x - 1)(x + 1)
Now let's move on to the next questions:
2. The discontinuities in the function can occur when the denominator becomes zero. Therefore, we need to find the values of x that make (x - 1)(x + 1) = 0. Setting each factor equal to zero gives x = 1 and x = -1. These are the points where the function may have discontinuities.
The type of discontinuity at x = 1 and x = -1 depends on whether the numerator is also zero at these points. If the numerator is not zero at these points, we have vertical asymptotes. If the numerator is zero at these points, we have removable discontinuities or holes.
3. The domain of a function is the set of all possible x-values for which the function is defined. In this case, the function is defined for all real numbers except the values that make the denominator zero. So the domain of this function is all real numbers except x = 1 and x = -1.
4. To determine the range of the function, we need to consider the behavior of the function as x approaches positive or negative infinity. As x approaches positive or negative infinity, the denominator (4x^2 - 4) becomes very large, making the entire fraction close to zero. Therefore, the range of this function is all real numbers except 0. In interval notation, the range is (-∞, 0) U (0, +∞).