Question

Verify algebra radically if the function is even, odd, or neither.Need help with #7

Answer 1

(7). For a function to be even, then

`f(x)=f(-x)`

And if a function is odd, then

`f(x)=-f(-x)`

Now, let us check

`\begin{gathered} f(x)=x\sqrt[]{4-x^2} \\ f(-x)\text{ becomes , that is replacing }x\text{ with -}x \\ f(-x)=-x\sqrt[]{4-(-x)^2} \\ f(-x)=-x\sqrt[]{4-x^2} \\ So,\text{ }f(x)=x\sqrt[]{4-x^2}\text{ and }f(-x)=-x\sqrt[]{4-x^2}\text{ } \\ we\text{ can s}ee\text{ that }f(x)\\e f(-x) \end{gathered}`

Hence, the function is not even.

Let's check for odd

`\begin{gathered} \text{For odd} \\ f(x)=-f(-x) \\ f(x)=x\sqrt[]{4-x^2}\text{ } \\ -f(-x)=-(-x\sqrt[]{4-(-x)^2} \\ -f(-x)=-(-x\sqrt[]{4-x^2} \\ -f(-x)=x\sqrt[]{4-x^2} \\ \text{Now we can s}ee\text{ that }f(x)=x\sqrt[]{4-x^2}\text{ and }-f(-x)=x\sqrt[]{4-x^2} \\ So,\text{ }f(x)=-f(-x) \end{gathered}`

Hence the function is odd