Problem 107 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the domain of the function and identify any horizontal or vertical asymptotes. $$f(x)=\frac{x+1}{x^{2}+1}$$

Step-by-Step Solution

Answer

Domain of the function is (-∞,∞). No vertical asymptotes exist. The function has a horizontal asymptote at \(y = 0\).

Step 1: Finding the Domain

The denominator of the function \(f(x)\) is \(x^{2}+1\). Since the square of any real number is always non-negative, and in this case we are adding 1 to it, the denominator will always be greater than zero, hence no real value of x can make the denominator equal to zero. So, the domain of the function is \(-\infty < x < \infty \) or all real numbers.

Step 2: Finding Vertical Asymptotes

Vertical asymptotes occur at those values of x for which the value of the function approaches infinity. Since our function is defined for all real numbers, we have no vertical asymptotes in this function.

Step 3: Finding Horizontal Asymptotes

A horizontal asymptote of a function exists if the value of the function approaches some constant value as the input (x-value) approaches \(+\infty\) or \(-\infty\). To find the horizontal asymptotes (if they exist), we evaluate the limits of the function as x approaches \(+\infty\) and \(-\infty\). The limit of the function \(f(x) = \frac{x+1}{x^2+1}\) as \(x\rightarrow +\infty\) and \(x\rightarrow -\infty\) is 0. Therefore, the line \(y=0\) is a horizontal asymptote.

Key Concepts

Domain of a Function

When we talk about the domain of a function, we're referring to all the possible inputs, or 'x' values, that a given function can accept without leading to any undefined or non-real values. For the function

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Horizontal Asymptotes

Horizontal asymptotes are fascinating lines that show us where a function's value will 'settle' as the input grows very large or very small—in other words, as it heads toward text
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Vertical Asymptotes

Vertical asymptotes are straight lines parallel to the y-axis, and a function approaches these lines without ever touching them as the input reaches certain critical values. These are often points where the function is undefined, typically where a denominator equals zero in a rational function. Unlike horizontal asymptotes, which a function text
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Limits of a Function

The limit of a function is a fundamental concept in calculus that describes the behavior of a function as its argument gets close to a certain point. It can be thought of as the value that a function is approaching as its input gets infinitely close to a specific number. It doesn't necessarily mean the function text
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