even function

• Apr 25th 2013, 06:29 AM
kastamonu
even function
f(x)=2x^2+2mx+12
If f(x-3) is even, what is the value of "m"?
• Apr 25th 2013, 06:35 AM
Prove It
Re: even function
Do you mean f(x-3) is an even function?
• Apr 25th 2013, 07:49 AM
HallsofIvy
Re: even function
Quote:

Originally Posted by kastamonu
f(x)=2x^2+2mx+12
If f(x-3) is even, what is the value of "m"?

Did you consider just multiplying it out?
f(x- 3)= 2x^2- 12x+ 9+ 2mx- 6m+ 12= 2x^2+(2m- 12)x+ 21- 6m.

A polynomial function is even if and only if it has only even powers of x.
• Apr 25th 2013, 09:00 AM
kastamonu
Re: even function
Quote:

Originally Posted by Prove It
Do you mean f(x-3) is an even function?

Yes.
• Apr 25th 2013, 09:01 AM
kastamonu
Re: even function
Quote:

Originally Posted by HallsofIvy
Did you consider just multiplying it out?
f(x- 3)= 2x^2- 12x+ 9+ 2mx- 6m+ 12= 2x^2+(2m- 12)x+ 21- 6m.

A polynomial function is even if and only if it has only even powers of x.

Then (2m- 12)=0?
• Apr 25th 2013, 01:50 PM
HallsofIvy
Re: even function
Eventually, you will have to think for yourself!
• Apr 25th 2013, 04:52 PM
Prove It
Re: even function
Quote:

Originally Posted by HallsofIvy
Did you consider just multiplying it out?
f(x- 3)= 2x^2- 12x+ 9+ 2mx- 6m+ 12= 2x^2+(2m- 12)x+ 21- 6m.

A polynomial function is even if and only if it has only even powers of x.

That is not true. A function is even if \displaystyle \begin{align*} f(-x) = f(x) \end{align*} for all x.

Consider the function \displaystyle \begin{align*} f(x) = (x - 1)^2 \end{align*}, then \displaystyle \begin{align*} f(-x) = (-x-1)^2 \neq (x-1)^2 \end{align*}.

The OP needs to check what f(x-3) is and then set it equal to f( -(x-3) ).
• Apr 25th 2013, 04:57 PM
HallsofIvy
Re: even function
Quote:

Originally Posted by Prove It
That is not true. A function is even if \displaystyle \begin{align*} f(-x) = f(x) \end{align*} for all x.

Consider the function \displaystyle \begin{align*} f(x) = (x - 1)^2 \end{align*}, then \displaystyle \begin{align*} f(-x) = (-x-1)^2 \neq (x-1)^2 \end{align*}.

Yes, and $(x- 1)^2= x^2- 2x+ 1$ has x to an odd power.

Quote:

The OP needs to check what f(x-3) is and then set it equal to f( -(x-3) ).
No, that would check if f is even. The OP needs to check if f(x- 3)= f(-x-3).
• Apr 25th 2013, 11:27 PM
kastamonu
Re: even function
f(x- 3)= f(-x-3)? Why do I have to check in this way?