2.6.3. Part (c): Injectivity of g

Try these first

1. 1.

   What does it mean for a function to be injective?

   Hint

   Think about one-to-one relationships.

2. 2.

   Is the function f(x) = x^2 injective over the real numbers?

   Hint

   Check distinct inputs leading to the same output.

3. 3.

   What guarantees a function to be injective?

   • Unique output for each input
   • Multiple outputs for one input
   • Every output is covered

   Hint

   Define injectivity clearly.

4. 4.

   If g∘f is injective, does that imply g is injective?

   • True
   • False

   Hint

   Consider the definitions discussed.

5. 5.

   Given two functions f(x) = x + 2 and g(x) = x^2, discuss and prove if g∘f is injective.

   Hint

   Analyze the composition step by step.

6. 6.

   Construct a real-world situation illustrating a surjective function that fails to be injective.

   Hint

   Think of various qualifications leading to common roles.