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QuestionFind the difference quotient for the function f(x)=−4x+7f(x)=-4x+7: f(x+h)−f(x)h,h≠0\frac{f(x+h)-f(x)}{h}, h \neq 0. Simplify your answer.

Studdy Solution

STEP 1

Assumptions1. The function is given as f(x)=−4x+7f(x)=-4x+7 . We are asked to find the difference quotient, which is f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}

STEP 2

First, we need to find f(x+h)f(x+h). This is done by replacing every instance of xx in the function with (x+h)(x+h).
f(x+h)=−4(x+h)+7f(x+h) = -4(x+h) +7

STEP 3

implify the expression f(x+h)f(x+h).
f(x+h)=−x−h+7f(x+h) = -x -h +7

STEP 4

Now, we can substitute f(x+h)f(x+h) and f(x)f(x) into the difference quotient formula.
f(x+h)−f(x)h=−4x−4h+7−(−4x+7)h\frac{f(x+h)-f(x)}{h} = \frac{-4x -4h +7 - (-4x +7)}{h}

STEP 5

implify the numerator of the fraction.
f(x+h)−f(x)h=−4hh\frac{f(x+h)-f(x)}{h} = \frac{-4h}{h}

STEP 6

Finally, simplify the fraction to find the difference quotient.
f(x+h)−f(x)h=−4\frac{f(x+h)-f(x)}{h} = -4The difference quotient of the function f(x)=−4x+f(x)=-4x+ is −4-4.

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