The expression in the image is:

$y^3(x - 9) = 1$

Finding the Domain

To find the domain of the function, we need to determine for which values of $x$ and $y$ the expression is defined.

• The expression $y^3(x - 9) = 1$ implies that the expression is defined for all $x$ and $y$ such that $x - 9 \neq 0$ (since $y^3 \neq 0$ for any real number $y$).
• Therefore, $x \neq 9$.

Thus, the domain of the function is all real numbers except $x = 9$: $\text{Domain: } x \in \mathbb{R}, \; x \neq 9.$

Finding the Range

To find the range, we solve the equation for $y$:

$y^3 = \frac{1}{x - 9}$

Since $x \neq 9$, $x - 9$ can take any non-zero real value. Therefore, $\frac{1}{x - 9}$ can also take any non-zero real value.

• Since $y^3 = \frac{1}{x - 9}$, $y^3$ can be any non-zero real number.
• $y$ itself can therefore be any non-zero real number.

Thus, the range of the function is all real numbers except $y = 0$: $\text{Range: } y \in \mathbb{R}, \; y \neq 0.$

Would you like more details, or do you have any questions?

Related Questions:

1. What happens to the graph of the function as $x$ approaches 9?
2. Can we rewrite this expression in terms of inverse functions?
3. How would you describe the behavior of $y$ as $x$ becomes very large or very small?
4. What are the asymptotes for this function, if any?
5. How would you plot this function on a coordinate plane?

Tip:

When determining the domain and range of a function, always consider the values that make the function undefined, such as division by zero or taking the square root of a negative number in the real number system.