help please with these 2 questions

odd and even funtions

given

$y=x^3\quad if \quad x\ge0\\ y=x \quad if \quad x<0\\\\~\\ f(2)=2^3=8 \qquad f(-2)=-2\\ f(-2)\ne f(2) \quad \\\mbox{therefore f(x) is not even}$

You can see that this graph is not symmetrical about the y axis so it is not even.

$f(x)=x^3\quad if \quad x\ge0\\ f(x)=x \quad if \quad x<0\\\\~\\ f(|x|) \quad \mbox{This results in the positive x part of the graph being reflected across the y axis. }\\ \mbox{For instance}\\ f(|2|)=f(2)=8 \qquad f(|-2|)=f(2)=8\\ \mbox{Since it is reflected across the y axis then it must be an even function.}\\$

$f(x)=x^3\quad if \quad x\ge0\\ f(x)=x \quad if \quad x<0\\\\~\\ | f(x)| \quad \mbox{This results in the NEGATIVE x part of the graph being reflected across the x axis. }\\ \mbox{For instance}\\ |f(2)|=|8|=8 \qquad | f(-2)|=|-2|=2\\ \mbox{Since it is NOT symmetrical about the y axis it is NOT even}\\$

Hi Brittany

I will answer better when I get home but this is what you need to know.

Even functions are symmetrical about the y axis. So f(x)=f(-x)

Odd functions have rotational symmetry of 180 degrees about the origin. So. f(-x)=-f(x)

So you can work these out by algebraic substitution or by looking at the graphs :)

Maybe that is enough of an answer. Let me know :))

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.