If f(x) is ODD will the following be true.
The easiest way to decide if these things MUST be true is to see if you can think of an example where it is NOT true.
Perhaps the most basic odd funtion is f(x)=x^3
I know it is odd because I can picture it in my mind and also because
\((-x)^3 = -(x)^3\)
Now
f(x)=x^3
f(|x|) will replace the -x function values with the +x values.
The means that the graph will change to the positive x side reflected in the y axis so it will not be odd
|f(x)| will leave the positive x side untouched but the negative side will be reflected over the x axis. (NOT the y axis)
In this case it will LOOK the same as the last graph. So it will NOT be odd either.
The last example y=f(x+1)= (x+1)^3
this just moves the graph one unit to the LEFT so it will not go be odd any more either,
So none of them will necessarily result in an odd function.