https://file.aiquickdraw.com/mathbot/file/1736880113318z1w5gtts.jpg

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

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### 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 \( c = -4 \).

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### 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: \( a = 1.5 \), \( b = 0.5 \).

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### 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:
   \[
   g^{-1}(x) = \log_{0.5}\left(\frac{x + 4}{1.5}\right)
   \]

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### 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 \( (-\infty, \infty) \).

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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!

The graph of an exponential function g(x) = a(b)^x + c is shown. Use the graph to determine: (A) the value of c and why, (B) whether a > 0 or a < 0 and if 0 < b < 1 or b > 1, and (C) the inverse function and its domain and range.

Explore how to analyze exponential functions, determine parameters from a graph, and find the inverse function. Includes solving for c, identifying growth or decay behavior, and domain and range of the inverse. Suitable for high school math learners.

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