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QuestionSimplify f(3+h)−f(3)f(3+h)-f(3) for f(x)=x2−12f(x)=x^{2}-12.

Studdy Solution

STEP 1

Assumptions1. The function is f(x)=x−12f(x) = x^{} -12 . We are asked to find and simplify the expression f(3+h)−f(3)f(3+h) - f(3)

STEP 2

First, we need to find the value of the function at +h+h. We can do this by substituting +h+h into the function.
f(+h)=(+h)2−12f(+h) = (+h)^{2} -12

STEP 3

Now, we will expand the square in the equation.
(3+h)2=32+2⋅3⋅h+h2(3+h)^{2} =3^{2} +2 \cdot3 \cdot h + h^{2}f(3+h)=32+2⋅3⋅h+h2−12f(3+h) =3^{2} +2 \cdot3 \cdot h + h^{2} -12

STEP 4

implify the equation.
f(3+h)=9+6h+h2−12f(3+h) =9 +6h + h^{2} -12f(3+h)=h2+6h−3f(3+h) = h^{2} +6h -3

STEP 5

Next, we need to find the value of the function at 33. We can do this by substituting 33 into the function.
f(3)=32−12f(3) =3^{2} -12

STEP 6

implify the equation.
f(3)=9−12f(3) =9 -12f(3)=−3f(3) = -3

STEP 7

Now, we can find the value of the expression f(3+h)−f(3)f(3+h) - f(3) by substituting the values we found for f(3+h)f(3+h) and f(3)f(3).
f(3+h)−f(3)=(h2+6h−3)−(−3)f(3+h) - f(3) = (h^{2} +6h -3) - (-3)

STEP 8

implify the equation.
f(3+h)−f(3)=h2+6h−3+3f(3+h) - f(3) = h^{2} +6h -3 +3f(3+h)−f(3)=h2+6hf(3+h) - f(3) = h^{2} +6hSo, the simplified expression for f(3+h)−f(3)f(3+h) - f(3) is h2+6hh^{2} +6h.

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