anonymous
  • anonymous
Use the following three steps to find and simplify the difference quotient of the function f(x)=\frac{1}{x-6}.
Mathematics
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anonymous
  • anonymous
Use the following three steps to find and simplify the difference quotient of the function f(x)=\frac{1}{x-6}.
Mathematics
chestercat
  • chestercat
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anonymous
  • anonymous
f(x) = 1/x-6
anonymous
  • anonymous
I believe I do the correct work but for the last two steps it says I am wrong.
anonymous
  • anonymous

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anonymous
  • anonymous
Jhannybean
  • Jhannybean
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 \(x+h\)
anonymous
  • anonymous
1/x-6 = 1/(x+h)-6
anonymous
  • anonymous
then i'm suppose to subtract it by f(x)
Jhannybean
  • Jhannybean
Alright, now we plug it in to the formula. And you are already given \(f(x)\).
anonymous
  • anonymous
1/(x+h)-6 - 1/x = -h-6/x(x+h)-6
Jhannybean
  • Jhannybean
\[\frac{f(x+h)-f(x)}{h} = \frac{\dfrac{1}{x+h-6}-\dfrac{1}{x-6}}{h}\]
Jhannybean
  • Jhannybean
When I find a common denominator, I multiply both the denominators of the fraction together.
Jhannybean
  • Jhannybean
\[=\frac{\dfrac{x-6 -(x+h-6)}{(x-6)(x+h-6)}}{h}\]
Jhannybean
  • Jhannybean
Now let's simplify both the numerator and denominator.
anonymous
  • anonymous
isn't the top suppose to be -h
anonymous
  • anonymous
-h/(x+h-6)(x-6)
anonymous
  • anonymous
then -h/(x+h-6)(x-6)/h
Jhannybean
  • Jhannybean
Yep, you're on track. That's what I got too. \[=\frac{\dfrac{x-6-x-h+6}{(x-6)(x+h-6)}}{h} = -\frac{\dfrac{h}{(x-6)(x+h-6)}}{h}\]
Jhannybean
  • Jhannybean
Now remember the rule of dividing fractions? \(\dfrac{\dfrac{a}{b}}{c} \iff \dfrac{a}{bc}\)
anonymous
  • anonymous
its -h^2/(x+h-6)(x-6)
Jhannybean
  • Jhannybean
We're going to apply this rule to our simplified function. We multiply \((x-6)(x+h-6)\) with \(h\).
Jhannybean
  • Jhannybean
Not exactly.
Jhannybean
  • Jhannybean
\[ -\frac{\dfrac{h}{(x-6)(x+h-6)}}{h} = -\frac{h}{h(x-6)(x+h-6)}\]Therefore both the h's cancel out.
Jhannybean
  • Jhannybean
\[-\frac{\cancel{h}}{\cancel{h}(x-6)(x+h-6)}\]
anonymous
  • anonymous
ahhh i understand
anonymous
  • anonymous
it says the bottom is not correct
anonymous
  • anonymous
nevermind
Jhannybean
  • Jhannybean
when you think of a fraction in the form \(\dfrac{\dfrac{a}{b}}{c}\) think of \(\dfrac{a}{b} \cdot \dfrac{1}{c} = \dfrac{a}{bc} \)
anonymous
  • anonymous
i forgot the - lol
Jhannybean
  • Jhannybean
Ahh, good good.

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