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Let's solve each problem step by step by analyzing the graphs and matching them to the correct exponential function.

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Problem 1

Graph Description:
- The graph is decreasing (as $ x $ increases, $ f(x) $ decreases).
- It passes through points:
- $ (-2, 9) $
- $ (-1, 3) $
- $ (0, 1) $

We know that for any exponential function $ f(x) = a^x $, when $ x = 0 $, $ f(0) = 1 $. This matches all options.

Now check the behavior:

Try $ f(x) = \left(\frac{1}{3}\right)^x $:
- $ f(-2) = \left(\frac{1}{3}\right)^{-2} = 3^2 = 9 $ ✔
- $ f(-1) = \left(\frac{1}{3}\right)^{-1} = 3 $ ✔
- $ f(0) = 1 $ ✔

So this matches perfectly.

✔ Answer: $ f(x) = \left(\frac{1}{3}\right)^x $

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Problem 2

Graph Description:
- Increasing function.
- Passes through:
- $ (0, 1) $
- $ (1, 2) $
- $ (2, 4) $

Check $ f(x) = 2^x $:
- $ f(0) = 2^0 = 1 $ ✔
- $ f(1) = 2^1 = 2 $ ✔
- $ f(2) = 2^2 = 4 $ ✔

Perfect match.

✔ Answer: $ f(x) = (2)^x $

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Problem 3

Graph Description:
- Increasing function.
- Passes through:
- $ (0, 1) $
- $ (1, 2) $
- $ (2, 4) $

Same as Problem 2!

Check $ f(x) = 2^x $:
- $ f(0) = 1 $
- $ f(1) = 2 $
- $ f(2) = 4 $

All match.

✔ Answer: $ f(x) = (2)^x $

(Note: The options list $ f(x) = (1)^x $ twice — likely a typo. But since $ (1)^x = 1 $, it’s constant. So clearly not correct.)

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Problem 4

Graph Description:
- Decreasing function.
- Passes through:
- $ (-2, 25) $
- $ (-1, 5) $
- $ (0, 1) $

Try $ f(x) = \left(\frac{1}{5}\right)^x $:
- $ f(-2) = \left(\frac{1}{5}\right)^{-2} = 5^2 = 25 $ ✔
- $ f(-1) = \left(\frac{1}{5}\right)^{-1} = 5 $ ✔
- $ f(0) = 1 $ ✔

Perfect.

Alternatively, test $ f(x) = \left(\frac{1}{10}\right)^x $:
- $ f(-2) = 10^2 = 100 $ → too big ✘

So only $ \left(\frac{1}{5}\right)^x $ fits.

✔ Answer: $ f(x) = \left(\frac{1}{5}\right)^x $

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✔ Final Answers:

1. $ f(x) = \left(\frac{1}{3}\right)^x $
2. $ f(x) = (2)^x $
3. $ f(x) = (2)^x $
4. $ f(x) = \left(\frac{1}{5}\right)^x $

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🔍 Explanation Summary:

- Exponential functions of the form $ f(x) = a^x $:
- If $ a > 1 $: increasing
- If $ 0 < a < 1 $: decreasing
- Always pass through $ (0, 1) $
- Use key points like $ x = -1, 0, 1 $ to test which base gives correct values.
- For decreasing graphs, look for bases like $ \frac{1}{3}, \frac{1}{5}, \frac{1}{6} $, etc.
- For increasing graphs, look for bases like $ 2, 3, 4 $, etc.

Let me know if you'd like a visual explanation or how to sketch these!