How do i determine if this function (x-4)(x+3)(2x-1) is even or odd?
if x = +1
(-3)(4)(1) = -12
if x = -1
(-5)(2)(-3) = +30
so f(-x) is NOT f(x) so not even
now is f(-x) = -f(x)
nope, 30 is not -(-12) so not odd
Can you do this in a way where you can just leave x as f(-x) instead of substituting it as 1?
Yes, but why bother ?
let x = a and then -a
and multiply the mess out and see if
f(-a) = f(a)
or if
f(-a) = - f(a)
I could have used x = 51375 or -51375
but no need
it has to work for ANY x to be odd or even
Okay, thanks for the clarification! It definitely is easier with your method, however my teacher wants it done in a certain way.
To determine whether a function is even or odd, we need to observe the symmetry of the function. Even functions are symmetric with respect to the y-axis, and odd functions are symmetric with respect to the origin (the point (0,0)).
To check whether the given function is even or odd, we need to apply the following rules:
1. For even functions: f(-x) = f(x)
2. For odd functions: f(-x) = -f(x)
Let's apply these rules to the given function: (x-4)(x+3)(2x-1).
1. For even functions:
Substitute -x for x in the function:
(-x-4)(-x+3)(2(-x)-1)
Expand and simplify:
(-x-4)*(-x+3)*(-2x-1)
(x+4)(x-3)(-2x-1)
Compare it with the original function:
(x-4)(x+3)(2x-1)
The function remains unchanged when we substitute -x for x. Therefore, the function (x-4)(x+3)(2x-1) is an even function.
2. For odd functions:
Substitute -x for x in the function:
(-x-4)(-x+3)(2(-x)-1)
Expand and simplify:
(-x-4)*(-x+3)*(-2x-1)
-1 * (x+4)(x-3)(2x+1)
Compare it with the opposite of the original function:
-1 * (x-4)(x+3)(2x-1)
The function becomes the opposite of the original function when we substitute -x for x. Therefore, the function (x-4)(x+3)(2x-1) is not an odd function.
To summarize:
The function (x-4)(x+3)(2x-1) is an even function.