Question

Group Activity In Exercises \(37-40\) , verify that the function is continuous and state its domain. Indicate which theorems you are using, and which functions you are assuming to be continuous. $$y=\left\\{\begin{array}{ll}{\frac{x^{2}-1}{x-1},} & {x \neq 1} \\ {2,} & {x=1}\end{array}\right.$$

Short answer

Yes, the function is continuous for all real numbers which also makes up its domain.

Step-by-step solution

Identify the Domain

By examining the function, it can be seen that it has two parts. The first part of the function \(\frac{x^{2}-1}{x-1}\) is defined for all \(x \neq 1\). The second part of the function is defined only for \(x = 1\). Thus, the domain of the function is all real numbers.

Check Continuity for \(x \neq 1\)

The first part of the function is a quotient of two functions that are known to be continuous since they are polynomials. According to the continuous function theorem, the quotient of two continuous functions \(f(x) / g(x)\) is continuous at all points where \(g(x) \neq 0\). Here, \(g(x)\) = \(x-1\), which is not equal to zero for all \(x \neq 1\), so it is continuous everywhere except at \(x = 1\).

Check Continuity for \(x = 1\)

Looking at the second part of the function which is \(y=2\) for \(x = 1\), it is a constant function and constant functions are always continuous. Thus, it is continuous at \(x = 1\).

Consolidate results

Since the function is continuous at every point of its domain, we can conclude that the function \(y\) is continuous for its entire domain which is all real numbers.