Problem 11 - Precalculus Essentials

Question

Find the domain of each function. $$g(x)=\frac{1}{x^{2}+1}-\frac{1}{x^{2}-1}$$

Step-by-Step Solution

Answer

The domain of the function \(g(x)=\frac{1}{x^{2}+1}-\frac{1}{x^{2}-1}\) is all real numbers except \(x=1\) and \(x=-1\).

Step 1: Identify possible undefined points

Set each denominator equal to zero and solve for x to determine potential points where the function may be undefined. In this case, we have two denominators: \(x^{2}+1\) and \(x^{2}-1\). So, (i) \(x^{2}+1=0\) which gives \(x^{2}=-1\). Since the square of a real number cannot be negative, there is no real solution to this equation. This means there are no real x-values that will make this fraction undefined. (ii) \(x^{2}-1=0\) which gives \(x^{2}=1\). Here, we have real solutions as \(x=1\) or \(x=-1\). These values would make the second fraction undefined, as division by zero is undefined in mathematics.

Step 2: Find the domain

The domain of the function is all real numbers except the values that make the function undefined. From Step 1, we know that the function is undefined at x=1 and x=-1. Therefore, the domain of this function is all real numbers with the exception of 1 and -1.