how to get the equation of a graph that is morphed

Question

If i had a graph that was y = e^x until one point

then after that point it got steeper than the first y =e^x , lets say it was a y = e^3x

then it changed again into maybe a downwards y = e^x

i know you can write it as y = / e^x when x is something

e^3x when x is something else

etc...

but you cant really specalute about figures above the graph

if i wanted to speculate about what the ffgure for y would be after x is a value that hasnt been graphed

how i can make it repeat itself and turn it into a general equation

so that if the graph did repeat like it has on the graph , i can put in any x value and get a y value that would be accurate

Answer 1

f(x) = e^x for x<2

f(x) = e^3x for 2<x<3

f(x) = e^4x for 3<x<4

is a step wise function

with discontinuities at x=2 and 3

as y jumps from e to e^2 or about 2.7 to 7.4, and

as y jumps from e^2 to e^3