Question

Consider the function below.
f(x)=(x+6)/(x+4)
Evaluate the difference quotient for the given function. Simplify your answer. f(x)-f(4)/(x-4)??

Answer 1

Consider the function below. f(x)=(x+6)/(x+4) Evaluate the difference quotient for the given function. Simplify your answer. f(x)-f(4)/(x-4)??



\[\large f(x)=\frac{x+6}{x+4}\]


\[\large f(4)=\frac{4+6}{4+4}\]


\[\large f(4)=\frac{10}{8}\]


\[\large f(4)=\frac{5}{4}\]

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\[\large \frac{f(x)-f(4)}{x-4}\]


\[\large \frac{\frac{x+6}{x+4}-\frac{5}{4}}{x-4}\]


\[\large \frac{\frac{4(x+6)}{4(x+4)}-\frac{5(x+4)}{4(x+4)}}{x-4}\]


\[\large \frac{\frac{4x+24}{4(x+4)}-\frac{5x+20}{4(x+4)}}{x-4}\]


\[\large \frac{\frac{4x+24-(5x+20)}{4(x+4)}}{x-4}\]


\[\large \frac{\frac{4x+24-5x-20}{4(x+4)}}{x-4}\]


\[\large \frac{\frac{-x+4}{4(x+4)}}{x-4}\]


\[\large \frac{-x+4}{4(x+4)}\times\frac{1}{x-4}\]


\[\large \frac{-x+4}{4(x+4)(x-4)}\]


\[\large \frac{-(x-4)}{4(x+4)(x-4)}\]


\[\large \frac{-\cancel{(x-4)}}{4(x+4)\cancel{(x-4)}}\]


\[\large \frac{-1}{4(x+4)}\]


\[\large -\frac{1}{4x+16}\]
So \[\large \frac{f(x)-f(4)}{x-4}=-\frac{1}{4x+16}\]