24.1.2. Injective Functions

Try these first

1. 1.

   What is an injective function?

   Hint

   Think about what happens if two inputs give the same output.

2. 2.

   Is the function f(x) = 2x injective?

   Hint

   You can test with any two different x values.

3. 3.

   Which of the following is true about injective functions?

   • f(a) = f(b) implies a = b
   • f(a) = f(b) can imply a ≠ b
   • f(a) can have multiple values for b

   Hint

   Remember about the unique mapping quality.

4. 4.

   Is the function f(x) = x² an injective function over the reals?

   • True
   • False

   Hint

   Consider different values that yield the same result.

5. 5.

   Demonstrate that the function f(x) = x^3 is injective over the real numbers.

   Hint

   Consider the properties of cubic functions.

6. 6.

   If a function f: A → B is injective, what can be said about its inverse f^−1: B → A?

   Hint

   Reflect on how injections relate to inverses.