Let's analyze the uploaded question and work through the problems step by step:

Problem A:

• The function given is $g(x) = a(b)^x + c$.
• To determine the value of $c$, note that $c$ represents the horizontal asymptote of the graph.
• Observing the graph, the asymptote is at $y = -4$, so: $c = -4$

Reason: The function levels off as $x \to \pm\infty$, and the horizontal asymptote is evident in the graph.

Problem B:

Part i: $a > 0$ or $a < 0$?

• Observing the graph, the curve decreases as $x$ increases (it has a downward trend).
• The decreasing behavior indicates that $a > 0$ and $0 < b < 1$.

Part ii: $b > 1$ or $0 < b < 1$?

• Since the graph is decreasing (exponential decay), $0 < b < 1$.

Part iii: Determine $a$ and $b$ using points from the graph:

1. Use the points $(0, 2.5)$ and $(1, -2)$ to solve for $a$ and $b$.

   From $g(0) = 2.5$: $g(0) = a(b)^0 + c \implies 2.5 = a(1) - 4 \implies a = 6.5$

   From $g(1) = -2$: $g(1) = a(b)^1 + c \implies -2 = 6.5(b) - 4 \implies 6.5b = 2 \implies b = \frac{2}{6.5} \approx 0.3077$

Thus, $a = 6.5$ and $b \approx 0.3077$.

Problem C:

The inverse of $g(x) = 6.5(0.3077)^x - 4$ is a logarithmic function:

1. Start with $y = 6.5(0.3077)^x - 4$.
2. Solve for $x$: $y + 4 = 6.5(0.3077)^x$ $\frac{y + 4}{6.5} = (0.3077)^x$ $x = \log_{0.3077}\left(\frac{y + 4}{6.5}\right)$

The inverse function is: $g^{-1}(x) = \log_{0.3077}\left(\frac{x + 4}{6.5}\right)$

Problem D:

Domain and Range of the Inverse Function:

• Domain: The original range of $g(x)$, which is $(-4, \infty)$.
• Range: The original domain of $g(x)$, which is $(-\infty, \infty)$.

Final Summary:

1. $c = -4$.
2. $a = 6.5$, $b \approx 0.3077$, $0 < b < 1$, $a > 0$.
3. $g^{-1}(x) = \log_{0.3077}\left(\frac{x + 4}{6.5}\right)$.
4. Domain of $g^{-1}(x)$: $(-4, \infty)$.
5. Range of $g^{-1}(x)$: $(-\infty, \infty)$.

Would you like further clarification on any part or details about logarithmic functions? Here are some follow-up questions to explore:

1. Why does $c$ represent the horizontal asymptote?
2. How does $b$ affect whether the exponential graph increases or decreases?
3. What happens to the inverse function if $b > 1$?
4. Can you explain the steps to find the domain and range of $g^{-1}(x)$ in more detail?
5. How do transformations affect the graph of exponential and logarithmic functions?

Tip: To confirm your work with logarithmic bases, you can rewrite the function as $\ln(x)/\ln(b)$ if your calculator lacks logarithms with arbitrary bases.