# In the table below, fill in as many y values as you can if you know that f is an even function, and g is an odd function?

## Enter the letter "n" if you cannot fill in a cell How would I start this problem?

Jul 28, 2018

Kindly refer to Explanation.

#### Explanation:

Since $f$ is even, $f \left(x\right) = f \left(- x\right)$.

$\therefore f \left(- 2\right) = f \left(2\right) = - 10 , \mathmr{and} ,$.

$f \left(- 1\right) = f \left(1\right) = - 7$.

Next, $g$ is odd, so, $g \left(- x\right) = - g \left(x\right)$.

$\therefore g \left(- 2\right) = - g \left(2\right) = - \left(- 7\right) = 7 , \mathmr{and} ,$

$- g \left(1\right) = g \left(- 1\right) = - 6 \Rightarrow g \left(1\right) = 6$.

Observe that, for any odd function like $g$ in question,

$\forall x , g \left(- x\right) = - g \left(x\right) \therefore g \left(- 0\right) = - g \left(0\right)$.

$\therefore g \left(- 0\right) + g \left(0\right) = 0 , i . e . , g \left(0\right) + g \left(0\right) = 0$.

$\therefore 2 g \left(0\right) = 0$.

$\therefore g \left(0\right) = 0$.

The rest are n.