Chapter 9.4, Problem 74E

To determine

To calculate:

The following function is even, odd or neither by checking it’s symmetry with respect to both axes and the origin:

  ${\left(x-2\right)}^2=\left(y+4\right)$

Answer to Problem 74E

  ${\left(x-2\right)}^2=\left(y+4\right)$ is not symmetric $x-$ axis and $y-$ axis. It is neither even or odd.

Explanation of Solution

Given information:

  ${\left(x-2\right)}^2=\left(y+4\right)$

Calculation:

The given function is:

  $\begin{array}{l}{\left(x-2\right)}^2=\left(y+4\right)\\ f\left(x\right)={\left(x-2\right)}^2+4\end{array}$

Now, replace the value $\left(x\right)$ by $\left(-x\right)$ in the above equation:

  $\begin{array}{l}f\left(x\right)={\left(-x-2\right)}^2+4\\ f\left(x\right)={\left(x+2\right)}^2+4\\ f\left(-x\right)\ne f\left(x\right)\\ f\left(-x\right)\ne -f\left(x\right)\end{array}$

Hence, the given function is neither symmetric about $x-$ axis and $y-$ axis. Therefore, the function is neither even nor odd.