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If $ f $ and $ g $ are both even functions, is $ f $ + $ g $ even? If $ f $ and $ g $ are both odd functions, is $ f $ + $ g $ odd? What if $ f $ is even and $ g $ is odd? Justify your answers.

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(i) If $f$ and $g$ are both even functions, then $f(-x)=f(x)$ and $g(-x)=g(x) .$ Now $(f+g)(-x)=f(-x)+g(-x)=f(x)+g(x)=(f+g)(x),$ so $f+g$ is an even function.(ii) If $f$ and $g$ are both odd functions, then $f(-x)=-f(x)$ and $g(-x)=-g(x) .$ Now $(f+g)(-x)=f(-x)+g(-x)=-f(x)+[-g(x)]=-[f(x)+g(x)]=-(f+g)(x),$ so $f+g$ is an odd function(iii) If $f$ is an even function and $g$ is an odd function, then $(f+g)(-x)=f(-x)+g(-x)=f(x)+[-g(x)]=f(x)-g(x)$which is not $(f+g)(x)$ nor $-(f+g)(x),$ so $f+g$ is neither even nor odd. (Exception: if $f$ is the zero function, then$f+g$ will be odd. If $g$ is the zero function, then $f+g$ will be even. .

03:55

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

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

University of Nottingham

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.

04:13

If $ f $ and $ g $ are bot…

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If $f$ and $g$ are both ev…

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Suppose $f$ and $g$ are bo…

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If $f$ is an odd function …

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$\begin{array}{l}{\text { …

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If $f$ is an even function…

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If F isn't even function, then we know that opposite X values have the same Y value, so f of X equals f of the opposite of X. If G is even then, opposite X values have the same y value. So we know that JIA Becks equals G of the opposite of X. So then what would have plus g b half of X plus g of X? Well, we could replace f of X with f of the opposite of X since they're equal. And then we could replace G of X with G of the opposite of X since they're equal. And so now we know that f of x plus g of X is equal to f of the opposite of X plus g of the opposite of X, which means that F plus G is also even. What if f Izod and G is odd? What about F plus G? So if F is odd than F of the opposite of X will equal the opposite of F of X, we could also say that the opposite of F of the opposite of X equals f of X. That's a mouthful. If she is odd same thing. G of the opposite of X is equal to the opposite of G of X. We multiply both sides by negative one. The opposite of G of the opposite of X is equal to G of X. So what about F Plus G? Half of eggs, plus G of X could be replaced with. We could replace F of X with the opposite of F of the opposite of X. And then we could replace G of X with the opposite of G of the opposite of X. And this If we want Teoh factor out, the negative would get the opposite of f of the opposite of X plus Jeep the opposite of X. So comparing this line and this line, we see that f plus G is odd. All right, so what if one of them is even in one of them? Is that if f is even then, f of X equals the equals f of the opposite of X? If G Assad, then g of X equals the opposite of G of the opposite of X. So what about F plus G half of X plus g of X? We could replace r F of X with f of the opposite of X. We could replace R G of X, with the opposite of G of the opposite of X. So we have f of the opposite of X minus G of the opposite of X that does not look equivalent to the original. That does not look like the opposite of the original, so it's neither on nor even.

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