How can you define different continuous functions #f(x)# and #g(x)# with #f(0) = g(0)# and the same average rate of change over the interval #[0, 2]# ?

1 Answer
George C.
Oct 2, 2017

Example:

#f(x) = x+1#

#g(x) = f(x) + x(x-2) = x^2-x+1#

Explanation:

The average rate of change is just #(Delta y)/(Delta x)# (i.e. the total change in #y# divided by the total change in #x#), so we just have to ensure that #f(x)# and #g(x)# take the same value at each of the endpoints of the interval.

If #f(x)# is any continuous function defined on the interval #[0, 2]#, then we can define:

#g(x) = f(x) + x(x-2)#

to get a function with the same average rate of change, since:

#g(0) - f(0) = 0#

and

#g(2) - f(2) = 0#

For example:

#f(x) = x+1#

#g(x) = f(x) + x(x-2) = x^2-x+1#

graph{(y-x-1)(y-x^2+x-1) = 0 [-3.957, 6.043, -0.72, 4.28]}

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