Question

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

Answer 1

f(x) = 1/x-6

Answer 2

I believe I do the correct work but for the last two steps it says I am wrong.

Answer 3

@ganeshie8

Answer 4

@Hero

Answer 5

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 \(x+h\)

Answer 6

1/x-6 = 1/(x+h)-6

Answer 7

then i'm suppose to subtract it by f(x)

Answer 8

Alright, now we plug it in to the formula. And you are already given \(f(x)\).

Answer 9

1/(x+h)-6 - 1/x = -h-6/x(x+h)-6

Answer 10

\[\frac{f(x+h)-f(x)}{h} = \frac{\dfrac{1}{x+h-6}-\dfrac{1}{x-6}}{h}\]

Answer 11

When I find a common denominator, I multiply both the denominators of the fraction together.

Answer 12

\[=\frac{\dfrac{x-6 -(x+h-6)}{(x-6)(x+h-6)}}{h}\]

Answer 13

Now let's simplify both the numerator and denominator.

Answer 14

isn't the top suppose to be -h

Answer 15

-h/(x+h-6)(x-6)

Answer 16

then -h/(x+h-6)(x-6)/h

Answer 17

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

Answer 18

Now remember the rule of dividing fractions? \(\dfrac{\dfrac{a}{b}}{c} \iff \dfrac{a}{bc}\)

Answer 19

its -h^2/(x+h-6)(x-6)

Answer 20

We're going to apply this rule to our simplified function. We multiply \((x-6)(x+h-6)\) with \(h\).

Answer 21

Not exactly.

Answer 22

\[ -\frac{\dfrac{h}{(x-6)(x+h-6)}}{h} = -\frac{h}{h(x-6)(x+h-6)}\]Therefore both the h's cancel out.

Answer 23

\[-\frac{\cancel{h}}{\cancel{h}(x-6)(x+h-6)}\]

Answer 24

ahhh i understand

Answer 25

it says the bottom is not correct

Answer 26

nevermind

Answer 27

when you think of a fraction in the form \(\dfrac{\dfrac{a}{b}}{c}\) think of \(\dfrac{a}{b} \cdot \dfrac{1}{c} = \dfrac{a}{bc} \)

Answer 28

i forgot the - lol

Answer 29

Ahh, good good.