Find the even and odd part of the function f(x)=3x^2-2x+1
To find the even part of a function, we need to find the function that remains unchanged when we replace x with -x. To find the odd part of a function, we need to find the function that changes sign when we replace x with -x.
For the function f(x) = 3x^2 - 2x + 1:
1. Even part:
Replacing x with -x, we get f(-x) = 3(-x)^2 - 2(-x) + 1
Simplifying, f(-x) = 3x^2 + 2x + 1
To find the even part, we take the average of f(x) and f(-x).
Even part: (f(x) + f(-x))/2 = (3x^2 - 2x + 1 + 3x^2 + 2x + 1)/2
= (6x^2 + 2)/2
= 3x^2 + 1
Therefore, the even part of f(x) = 3x^2 - 2x + 1 is 3x^2 + 1.
2. Odd part:
Replacing x with -x, we get f(-x) = 3(-x)^2 - 2(-x) + 1
Simplifying, f(-x) = 3x^2 + 2x + 1
To find the odd part, we subtract f(-x) from f(x).
Odd part: (f(x) - f(-x))/2 = (3x^2 - 2x + 1 - 3x^2 - 2x - 1)/2
= (-4x)/2
= -2x
Therefore, the odd part of f(x) = 3x^2 - 2x + 1 is -2x.
To find the even and odd parts of the function f(x) = 3x^2 - 2x + 1, we need to use the concept of symmetry.
1. Even Part:
The even part of a function is obtained by replacing x with -x. Let's calculate it:
f(-x) = 3(-x)^2 - 2(-x) + 1
= 3x^2 + 2x + 1
To find the even part, we take the average of f(x) and f(-x):
Even part = (f(x) + f(-x)) / 2
= (3x^2 - 2x + 1 + 3x^2 + 2x + 1) / 2
= (6x^2 + 2) / 2
= 3x^2 + 1
Therefore, the even part of the function f(x) = 3x^2 - 2x + 1 is 3x^2 + 1.
2. Odd Part:
The odd part of a function is obtained by subtracting the even part from the original function. Let's calculate it:
Odd part = f(x) - Even part
= (3x^2 - 2x + 1) - (3x^2 + 1)
= 3x^2 - 2x + 1 - 3x^2 - 1
= -2x
Therefore, the odd part of the function f(x) = 3x^2 - 2x + 1 is -2x.