Range of the Function y = f(x) = (3x - 5)/(x^2 - 1)

To determine the range of a function, we need to find the set of all possible values that the function can output. In this case, we are given the function y = f(x) = (3x - 5)/(x^2 - 1).

1. Identifying the Domain
Before determining the range, we need to first identify the domain of the function. The domain consists of all possible input 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.

Since the denominator is (x^2 - 1), it becomes zero when x = 1 or x = -1. Therefore, the domain of the function is all real numbers except x = 1 and x = -1.

2. Finding Vertical Asymptotes
To find the vertical asymptotes, we need to determine the values of x for which the function approaches positive or negative infinity. This occurs when the denominator of the function approaches zero.

When x approaches 1, the denominator (x^2 - 1) approaches zero from the positive side, while when x approaches -1, the denominator approaches zero from the negative side. Therefore, the vertical asymptotes of the function are x = 1 and x = -1.

3. Analyzing the Behavior of the Function
To understand the behavior of the function, we can consider the limits as x approaches positive or negative infinity. Taking the limit of the function as x approaches infinity, we have:

lim(x -> ∞) (3x - 5)/(x^2 - 1)

By dividing the numerator and denominator by x^2, we get:

lim(x -> ∞) (3/x - 5/x^2)/(1 - 1/x^2)

As x approaches infinity, the terms 5/x^2 and 1/x^2 become negligible. Hence, the limit simplifies to:

lim(x -> ∞) (3/x) / 1

This gives us the limit as x approaches infinity as 0. Similarly, by taking the limit as x approaches negative infinity, we also get 0.

4. Determining the Range
From the analysis above, we observe that the function approaches 0 as x approaches both positive and negative infinity. This means that the range of the function includes the value 0.

Additionally, since the function is continuous and has vertical asymptotes at x = 1 and x = -1, the function cannot take any values between these asymptotes. Therefore, the range of the function is all real numbers except 0.

In summary, the range of the function y = f(x) = (3x - 5)/(x^2 - 1) is (-∞, 0) U (0, +∞).

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