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Math Help - Difference Quotients

  1. #1
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    Difference Quotients

    I'm trying to find the Difference Quotient of


    F(x) = X / X + 1


    So far I've gotten to

    F(a) = a / a + 1

    F(a + h) = a + h / a + h + 1

    From here i'm totally lost Could someone please help me Thanks
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  2. #2
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    I'm guessing the function is f(x) = \frac{x}{x+1}.

    Then the difference quotient is given by

    \frac{f(x+h)-f(x)}{h} = \frac{\frac{x+h}{(x+h)+1}-\frac{x}{x+1}}{h} =\frac{\frac{x+h}{(x+h)+1}\frac{x+1}{x+1}-\frac{x}{x+1}\frac{(x+h)+1}{(x+h)+1}}{h}
    =\frac{\frac{x^2+x+hx+h}{x^2+x+hx+h+x+1}-\frac{x^2+hx+x}{x^2+hx+x+x+h+1}}{h} =\frac{\frac{x^2+x+hx+h-x^2-hx-x}{x^2+2x+hx+h+1}}{h}
    =\frac{h}{h(x^2+2x+hx+h+1)} = \frac{1}{x^2+2x+hx+h+1} = \boxed{\frac{1}{(x+1)^2+hx+h}}.

    You had the right idea. You just had to kept simplifying.
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  3. #3
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    Hello, Jonathan!

    Find the Difference Quotient of: . f(x) \:=\:\frac{x}{x+1}

    So far I've gotten to:

    f(a) \:= \:\frac{a}{a + 1}

    f(a + h) \:= \:\frac{a + h}{a + h + 1} . . . . Good!

    We want: . \frac{f(a+h) - f(a)}{h}

    The numerator is: . f(a+h) - f(a) \;=\;\frac{a+h}{a+h+1} - \frac{a}{a+1}


    Get a common denominator and subtract:

    . . \frac{a+h}{a+h+1}\cdot\frac{a+1}{a+1} + \frac{a}{a+1}\cdot\frac{a+h+1}{a+h+1} \;=\;\frac{(a+h)(a+1) - (a(a+h+1)}{(a+h+1)(a+1)}

    . . = \;\frac{a^2 + a +ah + h - a^2 - ah - a}{(a+h+1)(a+1)} \;=\;\frac{h}{(a+h+1)(a+1)}


    Then divide by h\!:\;\;\frac{f(a+h) - f(a)}{h} \;=\;\frac{1}{h}\cdot\frac{h}{(a+h+1)(a+1)} \;=\;\boxed{\frac{1}{(a+h+1)(a+1)}}

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