https://file.aiquickdraw.com/mathbot/file/1726626286085cu4a9gbd.jpg

To determine which function matches the graph, we analyze the given point on the graph \((1, 3)\) and check which of the functions satisfies this condition.

Given the point \((1, 3)\), when \(x = 1\), the value of \(f(x)\) should equal 3.

Let’s check each of the given functions:

- **A. \(f(x) = 2^x\)**  
  When \(x = 1\), \(f(1) = 2^1 = 2\).  
  This does not match the point \((1, 3)\).

- **B. \(f(x) = 2^{x+1}\)**  
  When \(x = 1\), \(f(1) = 2^{1+1} = 2^2 = 4\).  
  This also does not match \((1, 3)\).

- **C. \(f(x) = 2^x + 1\)**  
  When \(x = 1\), \(f(1) = 2^1 + 1 = 2 + 1 = 3\).  
  This matches the point \((1, 3)\).

- **D. \(f(x) = 3^x + 1\)**  
  When \(x = 1\), \(f(1) = 3^1 + 1 = 3 + 1 = 4\).  
  This does not match \((1, 3)\).

- **E. \(f(x) = 3^x\)**  
  When \(x = 1\), \(f(1) = 3^1 = 3\).  
  This matches \((1, 3)\), but we need to check the overall shape of the graph.

### Final Conclusion:
Both options C (\(f(x) = 2^x + 1\)) and E (\(f(x) = 3^x\)) satisfy the point \((1, 3)\), but the graph's general growth pattern is more consistent with the exponential function \(f(x) = 2^x + 1\), where the graph shows a vertical shift by 1 unit.

Thus, the correct function is **C. \(f(x) = 2^x + 1\)**.

Would you like more details or have any questions? Here are five related questions:

1. How do we verify if an exponential function matches a graph?
2. What is the general form of an exponential function?
3. How does a vertical shift affect the graph of a function?
4. How can we determine the growth rate from the shape of an exponential curve?
5. What is the significance of the base in an exponential function?

**Tip**: When matching functions to graphs, always check a few points and the general shape of the graph!

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Learn how to identify an exponential function from a graph with the point (1, 3). This problem covers key concepts such as graph interpretation, exponential growth, and vertical shifts. Suitable for students in Grades 9-11.

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