Please show step by step thank you For each function below

Please show step by step , thank you

For each function below, determine whether or not it is injective (one-to-one), surjective (onto). Fully justify your answers. F: R Times R rightarrow R defined by f(x, y) = 2x - y^2 g: R^+ Times R rightarrow R Times R defined by g(x, y) = (3x + 2y, x^2)

Solution

1a) Suppose x=1 and y =1, then f(x,y) = 2 - 1=1

      Now let x= 1 and y = -1, f(x,y) = 2- 1 = 1.

      Since 2 elements in domain maps into only in image/co-domain, f is not injective.

      It is also onto as 2x - y^2 is defined for every x, y -> R x R

b) The key here is x belongs to R⁺ or the set of positive real numbers, x² is injective over R⁺

     => 3x + 2y, x² is also injective.

Since the function g(x,y) is defined for the entrie interval in domain, g surjective. Thus g is a bijection.