# 1.7 Binary Relations

## In- ,Sur-, and Bijections

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
2. $$2x$$

Exercise 2
3. $$x^2$$

Exercise 3
$$x^2$$ 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
5. $$\sin x$$

Exercise 5
$$\sin x$$ 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
$$x \sin x$$ is not an injection, since $$0 \sin 0 = 2\pi \sin 2 \pi$$.
7. $$e^x$$

Exercise 7
$$e^x$$ is not a surjection, since $$e^x$$ is always positive for real values of $$x$$.