Determining algebraically if function is even/odd

[ssb]

An even function satisfies the property f(-x)= f(x)

Question: f(x) = (x-1)(x+3)(1+x)(3x-9)

Sub -x: (-x-1) (-x+3) (1+(-x)) (3(-x)-9)
Answer I get: (x+1)(x-3)(x-1)(3x+9)

Therefore f(-x) does not equal f(x).

But according to my book this function is even.

 

[HallsofIvy]

An even function satisfies the property f(-x)= f(x)

Question: f(x) = (x-1)(x+3)(1+x)(3x-9)

Sub -x: (-x-1) (-x+3) (1+(-x)) (3(-x)-9)
Answer I get: (x+1)(x-3)(x-1)(3x+9)

Therefore f(-x) does not equal f(x).

Yes, it does! First, (x- 1)(1+ x) becomes (-x- 1)(1- x) which, factoring a "-1" out of each becomes (x+ 1)(x- 1) exactly the same as you started with. I think you saw that. What you missed is that (x+ 3)(3x- 9)= 3(x+ 3)(x- 3) becomes 3(-x+ 3)(-x- 3) and then factoring a "-1" out of the last two terms gives 3(x- 3)(x+ 3), exactly what you started with.

But according to my book this function is even.

 

[ssb]

Yes, it does! First, (x- 1)(1+ x) becomes (-x- 1)(1- x) which, factoring a "-1" out of each becomes (x+ 1)(x- 1) exactly the same as you started with. I think you saw that. What you missed is that (x+ 3)(3x- 9)= 3(x+ 3)(x- 3) becomes 3(-x+ 3)(-x- 3) and then factoring a "-1" out of the last two terms gives 3(x- 3)(x+ 3), exactly what you started with.

Ah I got it now, but question

does x always have to be isolated?

For ex.

Q: y=(2x+4)(x-1)(x-3)

do I factor out the "2", so it becomes;

y=2(x+2)(x-1)(x-3)

then go on to check f(-x)

 

[GrannySmith]

Even functions will have y axis symmetry, while odd functions have origin symmetry.

In other words, plug in (-x) in for all (x) values. If the equation is equal to the starting equation, you have an even function.

To check for origin symmetry, substitue (-y) in for all (y values). Then substitute (-x) in for all (x values). If your new equation is equivalent to the one you started with, you have an odd function.