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 \(-\infty\) to \(\infty\).

Step-by-step solution

Identify the function's domain

The function \(f(x)=\frac{x}{{x^{2}+1}}\) is defined for all real values of \(x\), because \(x^{2}+1\) is always greater than zero for any real number \(x\). Therefore, there's no \(x\) that would make the denominator equal to zero. Therefore, the domain of the function is all real numbers.

Inspect the function for discontinuities

A function can only be discontinuous at a point within its domain. But as determined in the previous step, \(f(x)\) is defined for all real values of \(x\). Therefore, we are looking for discontinuities over the entire real line. It's crucial to note that \(f(x)\) would be discontinuous if the denominator \(x^{2}+1\) ever equaled zero. But, as stated above, \(x^{2}+1\) is always positive. Therefore, there are no points of discontinuity in this function.

State the intervals of continuity

Since the function is defined and has no points of discontinuity over the entire real line, it is continuous on the interval \(-\infty\) to \(\infty\).

Key concepts

Domain of a function

Every function has a domain. The domain of a function is the set of all possible input values (typically represented by \( x \)) that will yield a valid output. In other words, it represents all the real number values that \( x \) can take on. For some functions, the domain might be restricted due to mathematical operations that are undefined for certain inputs. For instance, a square root function cannot take negative inputs, and a rational function like \( \frac{1}{x} \) cannot have \( x \) equal to zero because division by zero is undefined. However, in the case of the function \( f(x)=\frac{x}{x^{2}+1} \), the domain is all real numbers. This is because \( x^2 + 1 \) is always positive regardless of the value of \( x \), so there are no restrictions on the values that \( x \) can take.

Real numbers

Real numbers encompass both rational and irrational numbers, forming a complete set of numbers that can represent distances along a continuous line. They include every number you can think of, whether they are fractions like \( \frac{1}{2} \), whole numbers like 7, or irrational numbers like \( \sqrt{2} \). When we talk about a function's domain or continuity, we generally talk about it in the context of real numbers unless specified otherwise. The function \( f(x)=\frac{x}{x^{2}+1} \) is evaluated in the real number domain. Because real numbers are continuous, any real number can be a valid input to this function, making the function seamless across its domain.

Intervals of continuity

Continuity of a function means that it has no breaks, holes, or jumps over a certain interval or over its entire domain. When a function can be drawn without lifting the pencil from the paper on a graph over its domain, it is considered continuous. For \( f(x)=\frac{x}{x^{2}+1} \), since the function is defined for all \( x \) in the real number set, it is continuous everywhere on the real line. Thus, the interval of continuity for this function is from \( -\infty \) to \( \infty \). There are no points at which \( f(x) \) is undefined, and no jumps or breaks in its graph. Therefore, the function is smoothly plotted without interruption over the entire real axis.