Question

Indicate whether the two functions are equal.

If the two functions are not equal, then give an element of the domain on which the two functions have different values.

$f:Z\to Z,$ where $f\left(x\right)=\left|x+y\right|$

$g:Z\to Z,$ where $g\left(x\right)=\left|x\right|+\left|y\right|$

Answer 1

Explanation:

Using triangle inequality

$$\begin{gathered} \left|x+y\right|<\left|x\right|+\left|y\right| \\ \Rightarrow f\left(x\right)<g\left(x\right) \\ \Rightarrow f\left(x\right)\ne g\left(x\right) \end{gathered}$$

Therefore, the function $f$ is not equal to function $g$ in the domain $\left(-\infty ,\infty \right)$.