The problem asks to identify which of the given functions are odd functions. Let's go through the functions step by step to check whether each one is odd.

Recall:

A function $f(x)$ is odd if: $f(-x) = -f(x)$ This means that if you replace $x$ with $-x$, the result must be the negative of the original function.

Now, let's analyze the functions:

1. $y = x - 2$

Substitute $-x$ for $x$: $f(-x) = -x - 2$

Clearly, $f(-x) \neq -(x - 2) = -x + 2$, so this function is not odd.

2. $y = x^3 + 5x$

Substitute $-x$ for $x$: $f(-x) = (-x)^3 + 5(-x) = -x^3 - 5x$

Now, check if this is the negative of the original function: $-(x^3 + 5x) = -x^3 - 5x$

Since $f(-x) = -f(x)$, this function is odd.

3. $y = \frac{x}{4}$

Substitute $-x$ for $x$: $f(-x) = \frac{-x}{4} = -\frac{x}{4}$

Check if this is the negative of the original function: $-\frac{x}{4} = -f(x)$

Thus, this function is odd.

4. $y = 4x^3 - x + 2$

Substitute $-x$ for $x$: $f(-x) = 4(-x)^3 - (-x) + 2 = -4x^3 + x + 2$

Clearly, $f(-x) \neq -(4x^3 - x + 2) = -4x^3 + x - 2$, so this function is not odd.

Conclusion:

Among the given functions, the odd functions are:

• $y = x^3 + 5x$
• $y = \frac{x}{4}$

Would you like further details or clarification on any step?

Here are 5 related questions to explore further:

1. What are the properties of even functions?
2. How can we graphically identify odd functions?
3. Can a function be both even and odd?
4. How does symmetry relate to odd functions?
5. Can a polynomial of even degree be an odd function?

Tip: Odd functions have rotational symmetry about the origin, which can help when sketching or analyzing graphs!