To find the $y$-intercept and the zeros of the function $f(x) = \sqrt[3]{x}$, let's follow these steps:

1. Finding the $y$-Intercept:

The $y$-intercept occurs when $x = 0$. Substituting $x = 0$ into the function:

$f(0) = \sqrt[3]{0} = 0$

So, the $y$-intercept is $(0, 0)$.

2. Finding the Zeros:

The zeros of the function occur when $f(x) = 0$. We set the function equal to zero and solve for $x$:

$\sqrt[3]{x} = 0$

To eliminate the cube root, cube both sides:

$x = 0$

So, the function has a zero at $x = 0$.

Summary:

• $y$-intercept: $(0, 0)$
• Zeros: $x = 0$

Would you like more details or have any questions?

Here are 5 related questions for further practice:

1. What is the domain of $f(x) = \sqrt[3]{x}$?
2. How does the graph of $f(x) = \sqrt[3]{x}$ compare to $f(x) = \sqrt{x}$?
3. What are the transformations of $f(x) = \sqrt[3]{x + 2}$?
4. What is the slope of the tangent line to $f(x) = \sqrt[3]{x}$ at $x = 1$?
5. Find the zeros of the function $g(x) = \sqrt[3]{x - 4}$.

Tip: Unlike square roots, cube roots can take any real number, including negative numbers, and they have no restrictions on their domain.