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Chapter 1: Problem 53

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

Short Answer

Expert verified
The function \( f(x) = \frac{x}{x^{2} + 1} \) is continuous on the interval \(-\infty, +\infty \)

Step by step solution

01

Understanding the function

The given function is a rational function and it takes the form \( f(x) = \frac{x}{x^{2} + 1} \), a simple form of a rational function as a ratio of two polynomial functions.
02

Identifying potential discontinuities

Rational functions typically have discontinuities where the denominator is equal to 0. In this case, our denominator is \(x^{2} + 1\). There is no real value of x that can make our denominator equal to 0. Hence, the function is defined for all real numbers.
03

Checking for other types of discontinuities

Other potential discontinuities could arise from some functional transformations, but there is no indication of such in our function. Therefore, our rational function is continuous everywhere in its domain since there are no places where the function goes undefined, or where a discontinuity can occur.
04

Concluding the intervals of continuity

Since there are no discontinuities in our function, we can safely state that our function is continuous everywhere. This can be described as the function being continuous on the interval \(-\infty, +\infty \)

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Most popular questions from this chapter

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