24.1.3. Surjective Functions

Try these first

1. 1.

   Define surjective function.

   Hint

   Think about how outputs relate to inputs.

2. 2.

   Is f(x) = x^2: ℤ -> ℤ+ surjective? Explain.

   Hint

   Consider the potential outputs when inputs are negative.

3. 3.

   What must be true for a function to be considered surjective?

   • Every element has multiple pre-images
   • Every element in the co-domain has at least one pre-image
   • No element in the co-domain can be mapped

   Hint

   Think about the purpose of a surjective function.

4. 4.

   True or False: The function f(x) = sin(x) from R to R is surjective.

   • True
   • False

   Hint

   What are the limits of the sine function?

5. 5.

   Design a surjective function from ℝ to ℝ and prove it remains surjective under defined conditions.

   Hint

   Define mappings for both linear and nonlinear functions in real number domains.

6. 6.

   Prove the non-surjectiveness of f(x) = sqrt(x) over integers to integers.

   Hint

   Consider outputs possible given the input constraints you set.