What are the excluded values of $x$ for $\frac{x+1}{x^2-1}$?
A rational expression of the form \(\frac{P(x)}{Q(x)}\) is undefined whenever the denominator \(Q(x)\) is equal to zero. To find the excluded values for the given expression \(\frac{x+1}{x^2-1}\), we must determine the values of \(x\) that make the denominator zero. Here, the denominator is \(D(x) = x^2 - 1\).
We set the denominator equal to zero to identify the excluded values:
\[x^2 - 1 = 0\]
We can factor the expression \(x^2 - 1\) using the difference of squares formula, \(a^2 - b^2 = (a-b)(a+b)\). Applying this, we get:
\[(x - 1)(x + 1) = 0\]
Now, we solve for \(x\) by setting each factor equal to zero:
These values, \(x = 1\) and \(x = -1\), make the denominator zero, which would cause division by zero. Therefore, these values must be excluded from the domain of the expression. Comparing this to the given options, the correct answer is \(x=-1, x=1\).
Excluded values appear in real-world scenarios whenever a formula involves division. For example, if you are calculating the average cost per item using the function \(C(x) = \frac{100}{x-5}\), where \(x\) is the number of items, you can immediately identify that \(x=5\) is an excluded value because it would imply dividing by zero (which makes no physical sense). Recognizing these points helps avoid errors in data modeling, engineering calculations, and financial projections.