1.7 Binary Relations

For each of the following real-valued functions on the real numbers \(\mathbb{R}\), indicate whether it is a bijection, a surjection but not a bijection, an injection but not a bijection, or neither an injection nor a surjection.

1. \(x+2\)

   Exercise 1

     a bijection a surjection but not a bijection an injection but not a bijection neither an injection nor a surjection a bijection 

2. \(2x \)

   Exercise 2

     a bijection a surjection but not a bijection an injection but not a bijection neither an injection nor a surjection a bijection 

3. \(x^2 \)

   Exercise 3

     a bijection a surjection but not a bijection an injection but not a bijection neither an injection nor a surjection neither an injection nor a surjection 
   \(x^2 \) is not a surjection, since negative numbers could not be squares of real numbers. \(x^2 \) also is not an injection, since \((-1)^2=1^2\).

4. \(x^3 \)

   Exercise 4

     a bijection a surjection but not a bijection an injection but not a bijection neither an injection nor a surjection a bijection 

5. \(\sin x \)

   Exercise 5

     a bijection a surjection but not a bijection an injection but not a bijection neither an injection nor a surjection neither an injection nor a surjection 
   \(\sin x \) is not a surjection, since \(-1 \leq \sin x \leq 1 \). \(\sin x \) is not an injection, since \(\sin 0 = \sin 2\pi \).

6. \(x \sin x \)

   Exercise 6

     a bijection a surjection but not a bijection an injection but not a bijection neither an injection nor a surjection a surjection but not a bijection 
   \(x \sin x \) is not an injection, since \(0 \sin 0 = 2\pi \sin 2 \pi \).

7. \(e^x \)

   Exercise 7

     a bijection a surjection but not a bijection an injection but not a bijection neither an injection nor a surjection an injection but not a bijection 
   \(e^x \) is not a surjection, since \(e^x \) is always positive for real values of \(x \).
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