Question

Describe the interval(s) on which the function is continuous. $$ f(x)=\frac{x}{x^{2}+1} $$

Short answer

The function \(f(x)=\frac{x}{x^{2}+1}\) is continuous on the interval \(-\infty < x < +\infty\) or in other words, all real numbers.

Step-by-step solution

Title – Identify the Denominator

Based on the definition of the function \(f(x) = \frac{x}{x^{2}+1}\), identify the denominator which is \(x^{2}+1\).

Title – Determine where the denominator equals zero

Next, equate the denominator \(x^{2}+1\) to zero and solve for \(x\). Doing this would help find any points that may cause the function to be undefined due to division by zero. However, in our case, \(x^{2}+1 = 0\) has no real solution since \(x^{2}\) is always greater than or equal to zero for all real numbers. Therefore, the sum of \(x^{2}\) and 1 can never be equal to zero.

Title – Determine the intervals of continuity

The function \(f(x) = \frac{x}{x^{2}+1}\) is continuous on the entire real line since there are no values of \(x\) for which the function would be undefined.