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Determine whether $ f $ is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually.

$ f(x) = \dfrac{x}{x^2 + 1} $

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Odd function

00:49

Jeffrey Payo

Calculus 1 / AB

Calculus 2 / BC

Calculus 3

Chapter 1

Functions and Models

Section 1

Four Ways to Represent a Function

Functions

Integration Techniques

Partial Derivatives

Functions of Several Variables

Missouri State University

Baylor University

University of Michigan - Ann Arbor

Boston College

Lectures

04:31

A multivariate function is a function whose value depends on several variables. In contrast, a univariate function is a function whose value depends on only one variable. A multivariate function is also called a multivariate expression, a multivariate polynomial, a multivariate series, or a multivariate function of several variables.

12:15

In calculus, partial derivatives are derivatives of a function with respect to one or more of its arguments, where the other arguments are treated as constants. Partial derivatives contrast with total derivatives, which are derivatives of the total function with respect to all of its arguments.

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Determine whether f is eve…

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Determine whether $f$ is e…

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Determine whether $ f $ is…

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Okay, so we have f of X equals X over X squared plus one. And we want to know if it's odd, even or neither. So remember that for odd functions, opposite X values give you opposite. Why values? And they have origin, symmetry and for even functions opposite X values give you the same. Why values? And so they have y axis symmetry. So let's investigate for our function. And let's figure out what f of the opposite of exes will it be the same or the opposite, or neither compared to the original. So what we need to do is substitute the opposite of X in for X in our function, and it will look like that. And then we can simplify. And we know that the opposite of X squared is just equivalent to X squared. So this looks like the opposite of our original function. Therefore, since half of the opposite of X equals the opposite of F of X, we have an odd function. So now when we look at the graph on the calculator, we should expect to see something that has origin symmetry. So we grab a calculator and type in the function and I'm going to try Zoom six for a standard negative 10 to 10 window. And here's how it looks. We could even change the window dimensions if we want. Maybe different y values. Let's say negative 3 to 3. Okay, that definitely looks like it has symmetry about the origin, so that confirms that it is an odd function.

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