SOLUTION: F(x+h)-f(x)/h where f(x)=x^2-x+4

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Question 1040330: F(x+h)-f(x)/h where f(x)=x^2-x+4
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Let's start by computing f(x+h)

f(x) = x^2 - x + 4
f(x+h) = (x+h)^2 - (x+h) + 4 ... replace every 'x' with (x+h)
f(x+h) = (x^2+2xh+h^2) - (x+h) + 4
f(x+h) = x^2+2xh+h^2 - x-h + 4

So f%28x%2Bh%29+=+x%5E2%2B2xh%2Bh%5E2+-+x-h+%2B+4
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Using what we found earlier, let's subtract f(x) from both sides

f%28x%2Bh%29+=+x%5E2%2B2xh%2Bh%5E2+-+x-h+%2B+4
f%28x%2Bh%29-f%28x%29+=+x%5E2%2B2xh%2Bh%5E2+-+x-h+%2B+4-f%28x%29
f%28x%2Bh%29-f%28x%29+=+x%5E2%2B2xh%2Bh%5E2+-+x-h+%2B+4-%28f%28x%29%29
f%28x%2Bh%29-f%28x%29+=+x%5E2%2B2xh%2Bh%5E2+-+x-h+%2B+4-%28x%5E2-x%2B4%29 Replace f(x) with x^2-x+4
f%28x%2Bh%29-f%28x%29+=+x%5E2%2B2xh%2Bh%5E2+-+x-h+%2B+4-x%5E2%2Bx-4
f%28x%2Bh%29-f%28x%29+=+2xh%2Bh%5E2-h
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Now that's divide that by h

f%28x%2Bh%29-f%28x%29+=+2xh%2Bh%5E2-h
%28f%28x%2Bh%29-f%28x%29%29%2Fh+=+%282xh%2Bh%5E2-h%29%2Fh
%28f%28x%2Bh%29-f%28x%29%29%2Fh+=+%28h%282x%2Bh-1%29%29%2Fh
%28f%28x%2Bh%29-f%28x%29%29%2Fh+=+2x%2Bh-1
----------------------------

So in the end, after simplifying, %28f%28x%2Bh%29-f%28x%29%29%2Fh is equal to 2x%2Bh-1