Identifying Where Functions Are Continuous

Determine where each function is continuous:

1. \(F(x) = x^{2}+5\) \(-\dfrac{x}{x^{2}+4}\)
2. \(G( x) =x^{3}+2x+\dfrac{x^{2}}{x^{2}-1}\)

Solution First we determine the domain of each function.

1. \(F\) is the difference of the two functions \(f( x) =x^{2}+5\) and \(g( x) =\dfrac{x}{x^{2}+4}\), each of whose domain is the set of all real numbers. So, the domain of \(F\) is the set of all real numbers. Since \(f\) and \(g\) are continuous on their domains, the difference function \(F\) is continuous on its domain.
2. \(G\) is the sum of the two functions \(f( x) =x^{3}+2x\), whose domain is the set of all real numbers, and \(g( x) =\dfrac{x^{2}}{x^{2}-1}\), whose domain is \(\left\{ x|x\neq -1,\hbox{ }x\neq 1\right\}\). Since \(f\) and \(g\) are continuous on their domains, \(G\) is continuous on its domain, \(\left\{ x|x\neq -1,\hbox{ }x\neq 1\right\}\).