The image shows the following exercise:

Exercise 3.2.16: Which of the following functions are surjective?

The options are:

(a) $f: \mathbb{R} \to \mathbb{R}$, where $f(x) = e^x$

(b) $f: (0, \infty) \to \mathbb{R}$, where $f(x) = \log x$

(c) $f: \mathbb{R} \to \mathbb{R}$, where $f(x) = \sin x$

Let's analyze each one for surjectivity:

1. (a) $f(x) = e^x$

   • Domain: $\mathbb{R}$
   • Codomain: $\mathbb{R}$
   • The exponential function $e^x$ has a range of $(0, \infty)$, meaning it only takes positive values. Therefore, it does not cover all real numbers (e.g., negative numbers are not in the range), so it is not surjective.

2. (b) $f(x) = \log x$

   • Domain: $(0, \infty)$
   • Codomain: $\mathbb{R}$
   • The logarithmic function $\log x$ is defined for positive real numbers and its range is $\mathbb{R}$, meaning for every real number $y$, there exists some $x \in (0, \infty)$ such that $\log x = y$. Hence, the function is surjective.

3. (c) $f(x) = \sin x$

   • Domain: $\mathbb{R}$
   • Codomain: $\mathbb{R}$
   • The sine function $\sin x$ has a range of $[-1, 1]$, meaning it does not cover all real numbers. Therefore, it is not surjective.

Conclusion:

• (a) is not surjective.
• (b) is surjective.
• (c) is not surjective.

Would you like further clarification or details on any of these concepts?

Here are 5 related questions to expand your understanding:

1. What is the definition of a surjective function (onto)?
2. How can you test whether a function is injective?
3. What is the range of the function $f(x) = \cos x$?
4. Can a function be both injective and surjective (bijective)?
5. How does the logarithmic function behave for inputs close to 0?

Tip: To check surjectivity, always compare the range of the function with the codomain. If they match, the function is surjective.