Let's analyze and solve the questions based on the image provided.

A. What is the value of $c$?

• Answer from the image: $c = -4$, because this is where the horizontal asymptote is located.
• Explanation: The horizontal asymptote of an exponential function $g(x) = a(b)^x + c$ corresponds to $c$. The graph levels off at $y = -4$, confirming that $c = -4$.

B. Analyze the graph:

1. Is $a > 0$ or $a < 0$?

   • From the graph, $a > 0$. This is because the graph demonstrates a downward decay as $x$ increases, consistent with positive $a$ for a decaying function.

2. Is $b > 1$ or $0 < b < 1$?

   • $0 < b < 1$. The graph shows exponential decay, which happens when the base $b$ is between 0 and 1.

3. Find values of $a$ and $b$:

   • Using $g(x) = a(b)^x + c$, and substituting the point $(1, -2.5)$: $-2.5 = a(b)^1 - 4$ Simplify: $-2.5 + 4 = ab$ $1.5 = ab$
   • Substituting the point $(0, -2.5)$: $-2.5 = a(1) - 4$ Solve for $a$: $a = 1.5$
   • Substituting $a = 1.5$ back into $ab = 1.5$: $1.5b = 1.5 \quad \Rightarrow \quad b = 1$

   Values: $a = 1.5$, $b = 0.5$.

C. Find the inverse function $g^{-1}(x)$:

The given function is: $g(x) = 1.5(0.5)^x - 4$

1. Replace $g(x)$ with $y$: $y = 1.5(0.5)^x - 4$
2. Solve for $x$ in terms of $y$: $y + 4 = 1.5(0.5)^x$ $\frac{y + 4}{1.5} = (0.5)^x$ Take the logarithm (base $0.5$) of both sides: $x = \log_{0.5}\left(\frac{y + 4}{1.5}\right)$
3. Swap $x$ and $y$ to get the inverse: $g^{-1}(x) = \log_{0.5}\left(\frac{x + 4}{1.5}\right)$

D. Domain and Range of $g^{-1}(x)$:

1. Domain: The original function $g(x)$ has a range of $(-4, \infty)$. Thus, the inverse has a domain of $(-4, \infty)$.

2. Range: The domain of $g(x)$ is $(-\infty, \infty)$, so the range of $g^{-1}(x)$ is $(-\infty, \infty)$.

Would you like further clarification or a step-by-step breakdown of any part? Here are some related questions:

1. How is the horizontal asymptote determined from the graph?
2. Why does $b < 1$ lead to exponential decay?
3. What is the process of finding the inverse function of a logarithmic/exponential equation?
4. Can we calculate exact numerical values of $a$ and $b$ if additional points are provided?
5. What are the implications of domain and range changes in inverse functions?

Tip: When solving for an inverse function, always ensure to check the domain and range constraints of the original function!