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!