Help! Want 10 points? Explain this to me because we have a quiz on Monday.?

Question

This is about odd and even functions. How do I know if an equation is odd or even or neither?

Right, if even, f(-x)=f(x), and if odd, f(-x)= -f(x)

What exactly does that mean?

Can you please give the complete solutions to these 2 items and briefly and clearly explain your answer. We have a quiz on Monday and I still don't get this lesson.

1. f(x) = 3x^3+2x
2.f(x) = IxI

Also, another question.

How do you find the domain of I3x-2I and how would it differ from 3x-2?

Whoever will make me end my frustration of not understanding this shall be chosen to have the best answer.

Answer 1

In order to determine that a given function f(x) is odd or even, we need to satisfy the conditions.



A function such that f(x) = f(-x) is called even function. Examples of even functions include 1 (or, in general, any constant function), IxI, cos x etc.



Let us take cosx for example.

f(x) = cosx

Now, f(-x) = cos(-x) = cosx = f(x)

=> f(x) = f(-x)

This means that even if, we put the negative of the dependent variable of a function the outcome will be the same as for the normal positive variable.



Similarly, an odd function is a function for which f(-x) = -f(x). Examples of odd functions include x, sinx, tanx etc.



Let us take sinx for example.

f(x) = sinx

f(-x) = sin(-x) = -sinx = -f(x)

=> -f(x) = f(-x).

This means that if, we take the negative of the dependent variable of a function, then the outcome will be negative of the whole function expression.



1) f(x) = 3x^3 + 2x

Now, let's check if it's an even or odd function.

Put f(-x);

f(-x) = 3(-x)^3 + 2(-x)

= -3x^3 -2x = -(3x^3 +2x)

= -f(x)

As f(-x) = -f(x), it's an odd function.



2) f(x) = IxI

= ( x, x>o)

( 0, x=0)

( -x, x<0)

Mod will always give a positive value whether you put +x or -x, hence, it's an even function ( f(x) = f(-x) ).



3) y = I3x-2I = ( 3x-2, 3x-2>0 = x>2/3)

( -(3x-2) = 2-3x, x<2/3)

( 0, x = 2/3 )



I hope it helps..

Answer 2

The "easiest" way to understand, and also the best concept for understanding the importance of even and odd functions is understanding the graph of the function. Draw the function on your graph paper.



After drawing it, if you fold your paper in half on the y-axis (left and right side), and the picture lines up on both sides the same -- it is said to be an "EVEN" function. By equation, it just means whenever you plug in a "negative of a number" or a "positive of a number" into the function, you get the same answer.



For an "ODD" function, the drawing is reflected diagonally from "top left" to "bottom right" diagonally (at 45 degrees) through the origin. Again, if you can fold the graph here and the picture lines up, then it is an "ODD" function.



If you draw the function, and it isn't either of the two above, than it is neither EVEN or ODD.



--- For problem two above... this graph would look like a LETTER V when you graph it. Since it reflects on the y axis (" folding in half left and right) it is EVEN.



---For problem 1 above... you when you graph it, you will notice that it reflects diagonally. So it is odd.

Answer 3

by changing x to -x you will have

1.f(-x)=3(-x)^3+2(-x)

=-3x^3-2x= -(3x^3+2x)= -f(x) so this function is an odd one.

2.f(-x)=I-xI=IxI=f(x) so this function is even.



the domain of that function is R.all real numbers.because there is no limmitation for x.for example in this case f(x)=1/x,you can't have zero for x.cause the answer will be infinity, and infinity is not a definite number.

both 3x-2 and I3x-2I are the same in domains.both's domain is R.

Answer 4

As an example:

cos(x) = cos(-x) so cos(x) is an even function

sin(x) = -sin(-x) so sin(x) is an odd function



Now to get back to the two questions

f(x) is given

Find the value of f(-x)



1. f(-x) = 3(-x)^3 + 2(-x)

= -3x^3-2x

= -(3x^3+2x)

= -f(x)

This is an odd function



2. f(-x) = |-x|

= |x|

= f(x)

This is an even function



Domain of |3x-2| :

|3x-2| is defined for all values of x

The domain is (-infinity , +infinity)



Domain of (3x-2):

(3x-2) is defined for all values of x

The domain is |3x-2| is defined for all values of x

The domain is (-infinity , +infinity)



The domains for both functions are the same. The ranges would be different as the first function would have a range of [0, +infinity) and the second function would have a range of (-infinity , +infinity)

Answer 5

Calculate f(-x):



1. f(-x) = 3[(-x)^3] + 2(-x) = 3*(-1)^3*x^3 - 2x =

= 3*(-1)*x^3 + (-1)*2x = (-1) * (3x^3 + 2x) = -f(x)

Hence, f(x) is odd



2. f(-x) = |-x| = |x| = f(x), hence f(x) is even.



The domain of |3x - 2| is all the real numbers.



The difference between |3x - 2| and 3x-2 is



that if 3x-2<0 |3x - 2| = -(3x - 2)

otherwise |3x-2|=3x-2