From the image, we have a graph showing two curves. I will help determine the domain and range for the function described by the red curve on the right side.

Domain:

The domain represents all possible $x$-values for which the function is defined.

From the graph:

• The red curve starts from an asymptote at $x = 0$ (it never actually touches the line $x = 0$).
• The curve extends to the right indefinitely, meaning it covers all $x > 0$.

So, the domain of the function is: $(0, \infty)$ (The function is defined for all values of $x$ greater than 0.)

Range:

The range represents all possible $y$-values that the function can take.

From the graph:

• The red curve starts from negative infinity as $x$ approaches 0, and then rises indefinitely.
• The curve approaches $y = -10$, but does not cross or reach this line. So, the graph starts slightly above $y = -10$ and increases infinitely.

Thus, the range of the function is: $(-10, \infty)$

Conclusion:

• Domain: $(0, \infty)$
• Range: $(-10, \infty)$

Would you like further explanation on how the asymptotes or behavior of the function affect the domain and range? Here are some follow-up questions to explore:

1. What is the significance of horizontal and vertical asymptotes in determining domain and range?
2. How can we determine the behavior of a function near an asymptote?
3. Could this type of function represent a rational function? Why or why not?
4. How would the graph change if the vertical or horizontal asymptote were shifted?
5. How can limits help in determining the behavior near asymptotes?

Tip: Always check for asymptotes when determining the domain and range of a function!