The given functions f(x), f(|x|), and |f(x)| are not even functions option (E) is correct.
It is defined as a special type of relationship and they have a predefined domain and range according to the function.
[tex]\rm f(x)=\left \{ {{x^3 \ if \ x\geq 0} \\ \atop {x \ \ if \ x < 0}} \right.[/tex]
To check whether the given functions are even or not we check algebraically.
If f(-x) = f(x)
Then the function is even.
By using the above property for f(x):
f(-x) = -x (x when x<0 and put the value -x)
The above function is not even.
For f(|x|)
If x≥0 then f(|x|) ⇒ f(x) ⇒ x³
If x<0 then f(|x|) ⇒ f(-x) ⇒ -x
The above function are also not even.
For |f(x)|
[tex]\rm |f(x)|=\left \{ {{|x^3| \ if \ x\geq 0} \\ \atop {|x| \ \ if \ x < 0}} \right. \Rightarrow \rm f(x)=\left \{ {{x^3 \ if \ x\geq 0} \\ \atop {-x \ \ if \ x < 0}} \right.[/tex]
We can clearly see the above function are not also even.
Thus, the given functions f(x), f(|x|), and |f(x)| are not even functions.
Learn more about the function here:
brainly.com/question/5245372
