SOLUTION: Given that f(x)= x^3-x+1, find (f(h)-f(-h))/2h Thanks!!

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Question 824097: Given that f(x)= x^3-x+1, find (f(h)-f(-h))/2h
Thanks!!
Answer by jim_thompson5910(35256) (Show Source):
You can put this solution on YOUR website!
f(x)= x^3-x+1

f(h)= h^3-h+1 ... replace EVERY copy of x with h

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f(x)= x^3-x+1

f(-h)= (-h)^3-(-h)+1 ... replace EVERY copy of x with -h

f(-h)= -h^3-(-h)+1

f(-h)= -h^3+h+1

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f(h)-f(-h) = (h^3-h+1) - (-h^3+h+1)

f(h)-f(-h) = h^3-h+1 + h^3 - h - 1

f(h)-f(-h) = 2h^3-2h

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So...

(f(h)-f(-h))/2h

turns into

(2h^3-2h)/2h

then you factor out 2h in the numerator to get

(2h(h^2-1))/2h

and then cancel out the '2h' terms to get

h^2 - 1

and then you can optionally factor to get

(h-1)(h+1)

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So the answer is either h^2 - 1 or (h-1)(h+1)

They are equivalent. The second expression is the factored form of the first (using the difference of squares rule).