I have a question about writing two exponential growth functions, please help

Question

Write two exonential growth functions f(x) and g(x) such that f(x) < g(x) for x < 3 and f(x) > g(x) for x > 3

Answer 1

The problem has erroneously confused two concepts with different notations:

exponential function that has the form f(x) = a^x + b and

exponential growth that has the form y(t) = ae^kt.

Accurate solutions required accurately worded problems.

Answer 2

Vyapti -- one easy way to set this up (other than by specifying functions that have different forms over pieces of their domain! -- but that's really cheating in terms of the intent of this assignment) is to have the two functions cross at x=3. Then the one you want to grow faster at x>3, make y=exp(2x) [ for example!] and the other be y =  a * exp (x). Solve for the coefficient a so as to make these two equal at x=3, and you're done.

There are many (an infinite number of) other ways of setting a similar relationship up -- you can have the arguments of the exp portions in the two equations be arbitrarily involved expressions using the variable x. The expressions just have to be equal for x=3; then the one with a higher power of x, or the higher coefficient of the same highest power of x, may be larger for the region x>3, and lesser for the region x<3 (though this does depend on the detailed shape of these expression graphs, so you have to be a little careful, and check your result graphically!)