This image discusses the properties of functions such as oddness, evenness, and boundedness, and asks to analyze the functions $f(x) = |x|$ and $g(x) = x^3$.

Let me answer the questions:

1. $f(x) = |x|$

• Odd or even? $|x|$ is an even function because $|x| = |-x|$ for all $x$. This satisfies the definition of an even function $f(-x) = f(x)$.

• Bounded or unbounded? $|x|$ is unbounded because as $x \to \infty$ or $x \to -\infty$, $|x| \to \infty$.

2. $g(x) = x^3$

• Odd or even? $x^3$ is an odd function because $(-x)^3 = -(x^3)$. This satisfies the definition of an odd function $f(-x) = -f(x)$.

• Bounded or unbounded? $x^3$ is unbounded because as $x \to \infty$ or $x \to -\infty$, $x^3 \to \infty$ or $x^3 \to -\infty$, respectively.

Summary Table:

Would you like me to explain any of these concepts (odd/even functions or bounded/unbounded functions) in more detail?

5 Related Questions:

1. What are the graphical differences between even and odd functions?
2. Can a function be neither odd nor even? Provide examples.
3. How can you mathematically determine if a function is bounded or unbounded?
4. Are all trigonometric functions either odd or even? Why?
5. How does the boundedness of a function affect its limits at infinity?

Tip:

To test if a function is odd or even, substitute $-x$ into the function and compare it with $f(x)$ and $-f(x)$.