Piecewise linear function

Okay, so imagine you have a toy train that moves on a track. The train can only go straight, but the track is made up of different parts that slope up or down.

A piecewise linear function is kind of like that track. It’s a way to describe how something – like the height of the track – changes depending on where you are. But instead of a train moving along it, we’re talking about numbers that change based on the input.

Let’s say we have a piecewise linear function that looks like this:

f(x) = { 3x - 4 if x ≤ 2
{ 5x + 1 if x > 2

The function is split into two parts, depending on the input (the x value). If x is less than or equal to 2, we use the first part of the function: 3 times x, minus 4. If x is greater than 2, we use the second part of the function: 5 times x, plus 1.

So let’s plug in some numbers and see how this works.

If we put x = 1 into the function, we’re using the first part:

f(1) = 3(1) - 4 = -1

But if we put x = 3 into the function, we’re using the second part:

f(3) = 5(3) + 1 = 16

So the function changes how it works depending on the input value. Kind of like how the train track changes slopes depending on where you are.