Which piecewise relation defines a function?

h(x)

Explanation:

1) f(x) is not a function because there is an ambiguity for x = 4.

x = 4 belongs to both intervals - 2 ≤ x ≤ 4 and x ≥ 4

Then you find two different possible images for x: 0 and -(2)^2 = - 4.

That makes that f(x) be not a function.

2) similar thing happens with g(x)

as per the given relation the value of g(x) for x = 2 is 4 and 4+1 = 5. Which makes that g(x) be not a function.

3) j(x) is not a function because the image of x = -4 is -3(-4) = 12 and 3.

4) h(x) is a function, because there is not any ambiguity in its definition, for every x in its domain there is only one image h(x).