For the function, find the average rate of change of f from 1 to x: $$\frac { f ( x ) - f ( 1 ) } { x - 1 } , x \neq 1$$
-$$f ( x ) = \frac { 6 } { x + 5 }$$
A) $- \frac { 1 } { x + 5 }$
B) $\frac { 6 } { ( x - 1 ) ( x + 5 ) }$
C) $\frac { 6 } { x ( x + 5 ) }$
D) $\frac { 1 } { x + 5 }$

Question

Quizplus · Accepted Answer

The average rate of change of a function f from x1 to x2 is given by the formula:$$\frac { f ( x_2 ) - f ( x_1 ) } { x_2 - x_1 }$$In this case, we are given that f(x) = 6/(x+5) and we want to find the average rate of change of f from 1 to x. So, we plug in x1 = 1 and x2 = x into the formula:$$\frac { f ( x ) - f ( 1 ) } { x - 1 } = \frac { \frac { 6 } { x + 5 } - \frac { 6 } { 1 + 5 } } { x - 1 }$$Simplifying this expression, we get:$$\frac { f ( x ) - f ( 1 ) } { x - 1 } = \frac { \frac { 6 } { x + 5 } - \frac { 6 } { 6 } } { x - 1 }$$$$\frac { f ( x ) - f ( 1 ) } { x - 1 } = \frac { \frac { 6 - 6 ( x + 5 ) } { x + 5 } } { x - 1 }$$$$\frac { f ( x ) - f ( 1 ) } { x - 1 } = \frac { \frac { 6 - 6x - 30 } { x + 5 } } { x - 1 }$$$$\frac { f ( x ) - f ( 1 ) } { x - 1 } = \frac { \frac { -6x - 24 } { x + 5 } } { x - 1 }$$$$\frac { f ( x ) - f ( 1 ) } { x - 1 } = \frac { -6x - 24 } { ( x + 5 ) ( x - 1 ) }$$$$\frac { f ( x ) - f ( 1 ) } { x - 1 } = \frac { -6 ( x + 4 ) } { ( x + 5 ) ( x - 1 ) }$$$$\frac { f ( x ) - f ( 1 ) } { x - 1 } = \frac { -6 } { x + 5 }$$Therefore, the average rate of change of f from 1 to x is -6/(x+5).

For the Function, Find the Average Rate of Change of F