SOLUTION: I believe this the right category. I was working on my calculus homework online and this question really stumped me. Find f(a), f(a + h), and the difference quotient f(a +

Algebra ->  Expressions-with-variables -> SOLUTION: I believe this the right category. I was working on my calculus homework online and this question really stumped me. Find f(a), f(a + h), and the difference quotient f(a +       Log On


   

Question 895964: I believe this the right category. I was working on my calculus homework online and this question really stumped me.
Find
f(a), f(a + h),
and the difference quotient
f(a + h) − f(a)/h
,
where
h ≠ 0.
f(x)=4/x+9

f(a)= ?
f(a + h)= ?
f(a + h) − f(a)/h=?


Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I'm going to assume the function is f%28x%29+=+4%2F%28x%2B9%29

First we need f(a)

f%28x%29+=+4%2F%28x%2B9%29

f%28a%29+=+4%2F%28a%2B9%29 Replace every 'x' with 'a'

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Now we need f(a+h)

f%28x%29+=+4%2F%28x%2B9%29

f%28a%2Bh%29+=+4%2F%28a%2Bh%2B9%29 Replace every 'x' with "a+h"

f%28a%2Bh%29+=+4%2F%28a%2B9%2Bh%29

f%28a%2Bh%29+=+4%2F%28%28a%2B9%29%2Bh%29

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Now because both f(a) and f(a+h) have an "a+9" in the denominator, I'm going to let z = a+9

So f%28a%29+=+4%2F%28a%2B9%29 turns into f%28a%29+=+4%2Fz

f%28a%2Bh%29+=+4%2F%28%28a%2B9%29%2Bh%29 turns into f%28a%2Bh%29+=+4%2F%28z%2Bh%29

this will make the work shown below a bit more easier
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Subtract the two. We do so by finding the LCD (in this case z(z+h)) and then getting each fraction to have this LCD

f%28a%2Bh%29+-+f%28a%29+=+4%2F%28z%2Bh%29+-+4%2Fz

f%28a%2Bh%29+-+f%28a%29+=+%284z%29%2F%28z%28z%2Bh%29%29+-+4%2Fz Multiply top and bottom of the first fraction by 'z'

Multiply top and bottom of the second fraction by 'z+h'

f%28a%2Bh%29+-+f%28a%29+=+%284z%29%2F%28z%28z%2Bh%29%29+-+%284z%2B4h%29%2F%28z%28z%2Bh%29%29

f%28a%2Bh%29+-+f%28a%29+=+%284z+-+%284z%2B4h%29%29%2F%28z%28z%2Bh%29%29

f%28a%2Bh%29+-+f%28a%29+=+%284z+-+4z-4h%29%2F%28z%28z%2Bh%29%29

f%28a%2Bh%29+-+f%28a%29+=+-%284h%29%2F%28z%28z%2Bh%29%29

f%28a%2Bh%29+-+f%28a%29+=+-%284h%29%2F%28%28a%2B9%29%28a%2B9%2Bh%29%29 Replace every 'z' with 'a+9'

--------------------------------------------------------------

The last step is to divide all of that by 'h'. Put another way, we multiply all of that by 1%2Fh. This is because dividing is the same as multiplying by the reciprocal.

f%28a%2Bh%29+-+f%28a%29+=+-%284h%29%2F%28%28a%2B9%29%28a%2B9%2Bh%29%29

Multiply both sides by 1%2Fh

We have these common 'h' terms (in top and bottom)

which divide to 1 and cancel out (they go away)

%28f%28a%2Bh%29+-+f%28a%29%29%2Fh+=+-4%2F%28%28a%2B9%29%28a%2B9%2Bh%29%29%29

which is the final answer. You could expand out the denominator, but that makes things messier than they have to be.