The problem describes an exponential function with the following characteristics:

1. Domain: All real numbers.
2. Range: All real numbers greater than -3.
3. Horizontal asymptote: $y = -3$.

Explanation:

• The horizontal asymptote $y = -3$ indicates that the graph approaches -3 but never touches it.
• Since the range is $y > -3$, the graph is shifted downward from the standard exponential graph and is above $y = -3$.
• The domain being all real numbers suggests the graph extends infinitely in both the positive and negative $x$-directions.
• The graph likely represents a function of the form $y = a \cdot b^x - 3$, where $a > 0$ and $b > 0$, because it has a positive growth trend.

Matching Graphs:

From the image, it appears the first graph (top) correctly fits the description:

• The graph is above $y = -3$.
• It has a horizontal asymptote at $y = -3$.
• It extends in both directions, consistent with the domain of all real numbers.

The second graph (bottom) does not fit, as it appears to have a horizontal asymptote at $y = 0$, which contradicts the problem's conditions.

Do you want further clarification or need help with related concepts?

Related Questions:

1. What is the general form of an exponential function with a horizontal shift?
2. How does changing the base $b$ of an exponential function affect its graph?
3. What is the significance of the range and asymptotes in defining a function?
4. How can transformations (shifts, stretches, etc.) of graphs be identified algebraically?
5. How is the horizontal asymptote calculated for exponential functions?

Tip:

Remember, the horizontal asymptote of a function $f(x) = a \cdot b^x + c$ is always $y = c$.