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Math Help - Even, Odd, or Neither

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    Even, Odd, or Neither

    Determine whether each of the following functions are even, odd, or neither. Please state the symmetry.

    a)f(x)=1 / (x^2+1) b)g(x)=x^2 / (1+x^4) c)h(x)= (2x^4) + (3x^2)
    d)f(x)={1 / (x^3+x)}^5 e)g(x)=x+(1/x) f)h(x)=x-(x^2)
    g)f(x)=2^x h)g(x)=(logx^2) / (log3x^4)

    For each of the above functions from a) through h) how do you determine if it is even,odd,or neither??
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by johett View Post
    Determine whether each of the following functions are even, odd, or neither. Please state the symmetry.

    a)f(x)=1 / (x^2+1) b)g(x)=x^2 / (1+x^4) c)h(x)= (2x^4) + (3x^2)
    d)f(x)={1 / (x^3+x)}^5 e)g(x)=x+(1/x) f)h(x)=x-(x^2)
    g)f(x)=2^x h)g(x)=(logx^2) / (log3x^4)

    For each of the above functions from a) through h) how do you determine if it is even,odd,or neither??
    a function is even if f(x) = f(-x). that is, if you replace x with -x and simplify, you get exactly the original function

    a function is odd if f(-x) = -f(x). that is, if you replace x with -x and simplify, you get the negative of the original function

    see the "Ways to test for symmetry" section here on how to test for other kinds of symmetry. i don't know what kinds you are expected to look for.

    note: symmetry about the y-axis is the same as being even, and symmetry about the origin is the same as being odd
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    can someone show me an example like question a) so I can have a better understanding thanks
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by johett View Post
    can someone show me an example like question a) so I can have a better understanding thanks
    i'll give you other examples. then try your questions and post your solutions and we'll tell you if you applied the rules correctly.

    example 1:


    the function f(x) = x^2 + 1 is even, since replacing x with -x we get:

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

    since f(-x) = f(x), the function is even


    example 2:

    the function f(x) = x^3 is odd, since if we replace x with -x we get:

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

    since f(-x) = -f(x), the function is odd


    example 3:

    the function f(x) = x^3 + 3 is neither even nor odd, since if we replace x with -x we get:

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

    clearly f(-x) \ne f(x) and f(-x) \ne -f(x). so the function is neither even nor odd


    This is my 59th post!!!!
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by johett View Post
    can someone show me an example like question a) so I can have a better understanding thanks
    Quote Originally Posted by johett View Post
    Determine whether each of the following functions are even, odd, or neither. Please state the symmetry.

    a)f(x)=1 / (x^2+1)
    d)f(x)={1 / (x^3+x)}^5
    f)h(x)=x-(x^2)

    For each of the above functions from a) through h) how do you determine if it is even,odd,or neither??
    a)
    f(x) = \frac{1}{x^2 + 1}

    f(-x) = \frac{1}{(-x)^2 + 1}

    f(-x) = \frac{1}{x^2 + 1} = f(x)
    so this function is even.

    d)
    f(x) = \left ( \frac{1}{x^3 + x} \right ) ^5

    f(-x) = \left ( \frac{1}{(-x)^3 + (-x)} \right ) ^5

    f(-x) = \left ( \frac{1}{-x^3 - x} \right ) ^5

    f(-x) = \left ( \frac{-1}{x^3 + x} \right ) ^5

    f(-x) = (-1)^5 \left ( \frac{1}{x^3 + x} \right ) ^5

    f(-x) = - \left ( \frac{1}{x^3 + x} \right ) ^5 = -f(x)

    so this function is odd.

    f)
    h(x) = x - x^2

    h(-x) = (-x) - (-x)^2

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

    which is equal to neither h(x) nor -h(x). So this function is neither even, nor odd.

    -Dan
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