f(x) is defined to be ODD and g(x) is defined to be EVEN, complete table

[hunter]

I need some help here.

Given two functions f(x) and g(x), which are only defined at x=(-3,-2,-1,0,1,2,3) but not at any intermediate values. And f(x) is defined to be an ODD function and g(x) is defined to be EVEN function.

I need to complete the following table;

x	f(x)	g(x)	f+g	g-f	fg	f/g	g/f
-3	1	6
-2	0	-2
-1	-1	1
0	4	0
1
2
3

I've been pulling my hair out here working to figure this out. It's probably simple but after working a problem for days, I'm shot. Can somebody provide some guidance/help here? It would be much appreciated. Thanks.

Hunter

 

[tkhunny]

You should have definitions of Odd and Even? Do you have them?

Please write them and let's have a look.

 

[hunter]

The data in the table is the only information provided. ODD and EVEN functions are already known and defined outside of the problem. The issue is what are the functions that provide the values for f(x) and g(x) with the given values of X (-3,-2,-1,0,1,2,3).

 

[soroban]

Hello, hunter!

Given two functions \(\displaystyle f(x)\) and \(\displaystyle g(x)\), which are defined at \(\displaystyle x=\{\text{-}3,\text{-}2,\text{-}1,0,1,2,3\}\) only.
And f(x) is defined to be an ODD function and g(x) is defined to be EVEN function.

I need to complete the following table;

\(\displaystyle \begin{array}{c|c|c||c|c|c|c|c|}x & f(x) & g(x) & f+g & g-f & fg & f/g & g/f \\ \hline \text{-}3 & 1 & 6 \\ \text{-}2 & 0 & \text{-}2\\ \text{-}1 & \text{-}1 & 1 \\ 0 & 4 & 0 \\ 1 \\ 2 \\ 3\\ \hline \end{array}\)

An ODD function is defined as: for all \(\displaystyle x,\:f(\text{-}x) \,=\,-f(x).\)
. . Baby talk: If you change the sign of \(\displaystyle x\), you change the sign of \(\displaystyle f(x).\)

An EVEN function is defined as: for all \(\displaystyle x,\:f(\text{-}x) \,=\,f(x)\)
. . Baby talk: If you change the sign of \(\displaystyle x,\:f(x)\) is unchanged.


Knowing this, you can complete the next two columns:

\(\displaystyle \begin{array}{c|c|c ||c|c|c|c|c|}x & f(x) & g(x) & f+g & g-f & fg & f/g & g/f \\ \hline \text{-}3 & 1 & 6 \\ \text{-}2 & 0 & \text{-}2\\ \text{-}1 & \text{-}1 & 1 \\ 0 & 4 & 0 \\ 1 & {\color{red}1} & {\color{red}1} \\ 2 & {\color{red}0} & {\color{red}{\text{-}2}} \\ 3 & {\color{red}{\text{-}1}} & {\color{red}{6}} \\ \hline \end{array}\)


Now you can complete the table:

\(\displaystyle \begin{array}{c|c|c|c|c|c|c|c|}x & f(x) & g(x) & f+g & g-f & fg & f/g & g/f \\ \hline \text{-}3 & 1 & 6 & {\color{blue}7} & {\color{blue}5} & {\color{blue}6} & {\color{blue}{\frac{1}{6}}} & {\color{blue}6} \\ \text{-}2 & 0 & \text{-}2\\ \text{-}1 & \text{-}1 & 1 \\
0 & 4 & 0 \\ 1 & 1 & 1 \\ 2 & 0 & \text{-}2 \\ 3 & \text{-}1 & 6 \\ \hline \end{array}\)

 

[hunter]

Thanks for your help and explanation. I was able to figure this out after stepping back and understanding the tests for ODD functions. It makes sense now. Don't know why I couldn't see it earlier.