Question

Sums of Even and Odd Functions If fand gare both even functions, is necessarily even? If both are odd, is their sum necessarily odd? What can you say about the sum if one is odd and one is even? In each case prove your answer.

Short answer

The sum of two even function is even. The sum of two odd function is odd. The sum of one odd and one even function is neither odd neither even function.

Step-by-step solution

Step 1. Answer the first question.

Let f and g be two even function and there sum is h.

$$\begin{gathered} h\left(x\right)=f\left(x\right)+g\left(x\right) \\ h(-x)=f(-x)+g(-x) \\ =f\left(x\right)+g\left(x\right) \end{gathered}$$ (Since f and g are even functions).

$=h\left(f\right(x\left)\right).$

Hence, there is sum is an even function.

Step 2. Answer the second question.

Let f and g be two odd function and there sum is h.

$\begin{array}{c}h\left(x\right)=f\left(x\right)+g\left(x\right)\\ h(-x)=f(-x)+g(-x)\end{array}$(Since f and g are even functions).

$\begin{array}{l}=-f\left(x\right)-g\left(x\right)\\ =-\left[f\left(x\right)+g\left(x\right)\right]\\ =-h\left(x\right).\end{array}$

Hence, the sum of two odd function is odd.

Step 3. Answer the third question.

Let f be odd function and g be even function and h be the sum of f and g.

$\begin{array}{c}h\left(x\right)=f\left(x\right)+g\left(x\right)\\ h(-x)=f(-x)+g(-x)\end{array}$

$=-f\left(x\right)+g\left(x\right)$

Hence, the sum of one odd and one even function is neither odd nor even.