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anonymous

  • one year ago

Use the following three steps to find and simplify the difference quotient of the function f(x)=\frac{1}{x-6}.

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  1. anonymous
    • one year ago
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    f(x) = 1/x-6

  2. anonymous
    • one year ago
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    I believe I do the correct work but for the last two steps it says I am wrong.

  3. anonymous
    • one year ago
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    @ganeshie8

  4. anonymous
    • one year ago
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    @Hero

  5. Jhannybean
    • one year ago
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    The difference quotient: \(\dfrac{f(x+h)-f(x)}{h}\) So let's begin by finding \(f(x+h)\) first. Wherever you see an x, replace it with \(x+h\)

  6. anonymous
    • one year ago
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    1/x-6 = 1/(x+h)-6

  7. anonymous
    • one year ago
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    then i'm suppose to subtract it by f(x)

  8. Jhannybean
    • one year ago
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    Alright, now we plug it in to the formula. And you are already given \(f(x)\).

  9. anonymous
    • one year ago
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    1/(x+h)-6 - 1/x = -h-6/x(x+h)-6

  10. Jhannybean
    • one year ago
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    \[\frac{f(x+h)-f(x)}{h} = \frac{\dfrac{1}{x+h-6}-\dfrac{1}{x-6}}{h}\]

  11. Jhannybean
    • one year ago
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    When I find a common denominator, I multiply both the denominators of the fraction together.

  12. Jhannybean
    • one year ago
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    \[=\frac{\dfrac{x-6 -(x+h-6)}{(x-6)(x+h-6)}}{h}\]

  13. Jhannybean
    • one year ago
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    Now let's simplify both the numerator and denominator.

  14. anonymous
    • one year ago
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    isn't the top suppose to be -h

  15. anonymous
    • one year ago
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    -h/(x+h-6)(x-6)

  16. anonymous
    • one year ago
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    then -h/(x+h-6)(x-6)/h

  17. Jhannybean
    • one year ago
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    Yep, you're on track. That's what I got too. \[=\frac{\dfrac{x-6-x-h+6}{(x-6)(x+h-6)}}{h} = -\frac{\dfrac{h}{(x-6)(x+h-6)}}{h}\]

  18. Jhannybean
    • one year ago
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    Now remember the rule of dividing fractions? \(\dfrac{\dfrac{a}{b}}{c} \iff \dfrac{a}{bc}\)

  19. anonymous
    • one year ago
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    its -h^2/(x+h-6)(x-6)

  20. Jhannybean
    • one year ago
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    We're going to apply this rule to our simplified function. We multiply \((x-6)(x+h-6)\) with \(h\).

  21. Jhannybean
    • one year ago
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    Not exactly.

  22. Jhannybean
    • one year ago
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    \[ -\frac{\dfrac{h}{(x-6)(x+h-6)}}{h} = -\frac{h}{h(x-6)(x+h-6)}\]Therefore both the h's cancel out.

  23. Jhannybean
    • one year ago
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    \[-\frac{\cancel{h}}{\cancel{h}(x-6)(x+h-6)}\]

  24. anonymous
    • one year ago
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    ahhh i understand

  25. anonymous
    • one year ago
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    it says the bottom is not correct

  26. anonymous
    • one year ago
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    nevermind

  27. Jhannybean
    • one year ago
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    when you think of a fraction in the form \(\dfrac{\dfrac{a}{b}}{c}\) think of \(\dfrac{a}{b} \cdot \dfrac{1}{c} = \dfrac{a}{bc} \)

  28. anonymous
    • one year ago
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    i forgot the - lol

  29. Jhannybean
    • one year ago
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    Ahh, good good.

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