Ican sort of understand the f(x)={x+1 part, but i don't understand the second halves, the -2x< 3 part. what's going on? where did they come from? ? why do they

This is the concept of a piecewise function. for your function, you have



the second halves tell you the interval in which the left equation is graphed in. they are defined as such. they are there because whoever made the notes/question/whatever the picture is literally defined the function to be like that. the graph ends up looking like that.

means between  (inclusive)  and  (inclusive). 

means between  (exclusive)  and  (inclusive). 

for example, between -2 and 3 (inclusive), the function graph will have the equation , which is a line with y-intercept of 1. then, between 3 (exclusive) and 7 (inclusive), the function graph will have equation  , which is a parabola with vertex at (5,0).

as for why those two definitions of the function can mean the same thing:



and 



and have the same value at  , so it does not matter which piece gets the "equal to." notice how they changed which piece gets the  (less than or equal to).

if both pieces were to be defined at  , then the function would not be a function (because it would technically have one domain element mapping to two range elements). so we have to leave one piece that doesn't include  .


i have attached another example of a piecewise function.

note that piecewise functions are not just limited to straight lines and parabolas; they can be anything that you can graph.