1. even function

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

2. Re: even function

Do you mean f(x-3) is an even function?

3. Re: even function

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.

4. Re: even function

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

5. Re: even function

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?

6. Re: even function

Eventually, you will have to think for yourself!

7. Re: even function

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 \displaystyle \begin{align*} f(-x) = f(x) \end{align*} for all x.

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

8. Re: even function

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

Consider the function \displaystyle \displaystyle \begin{align*} f(x) = (x - 1)^2 \end{align*}, then \displaystyle \displaystyle \begin{align*} f(-x) = (-x-1)^2 \neq (x-1)^2 \end{align*}.
Yes, and $\displaystyle (x- 1)^2= x^2- 2x+ 1$ has x to an odd power.

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).

9. Re: even function

f(x- 3)= f(-x-3)? Why do I have to check in this way?