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Answer :
- Evaluate $f(-1) = 5^{2(-1)} = 5^{-2} = \frac{1}{25}$.
- Evaluate $f(0) = 5^{2(0)} = 5^{0} = 1$.
- Evaluate $f(2) = 5^{2(2)} = 5^{4} = 625$.
- The values are $\boxed{\frac{1}{25}, 1, 625}$.
### Explanation
1. Understanding the problem
We are given the function $f(x) = 5^{2x}$ and we need to evaluate it at $x = -1, 0, 2$. This involves substituting these values into the function and simplifying.
2. Calculating f(-1)
First, let's evaluate $f(-1)$. We have $f(-1) = 5^{2(-1)} = 5^{-2}$. Since $5^{-2} = \frac{1}{5^2} = \frac{1}{25}$, we get $f(-1) = \frac{1}{25}$.
3. Calculating f(0)
Next, let's evaluate $f(0)$. We have $f(0) = 5^{2(0)} = 5^{0}$. Any non-zero number raised to the power of 0 is 1, so $f(0) = 1$.
4. Calculating f(2)
Finally, let's evaluate $f(2)$. We have $f(2) = 5^{2(2)} = 5^{4}$. Since $5^4 = 5 \times 5 \times 5 \times 5 = 625$, we get $f(2) = 625$.
5. Final Answer
Therefore, $f(-1) = \frac{1}{25}$, $f(0) = 1$, and $f(2) = 625$.
### Examples
Exponential functions are used to model various real-world phenomena, such as population growth, radioactive decay, and compound interest. For example, if a bacteria population doubles every hour, the population size can be modeled by an exponential function. Similarly, the amount of money in a bank account with compound interest grows exponentially over time.
- Evaluate $f(0) = 5^{2(0)} = 5^{0} = 1$.
- Evaluate $f(2) = 5^{2(2)} = 5^{4} = 625$.
- The values are $\boxed{\frac{1}{25}, 1, 625}$.
### Explanation
1. Understanding the problem
We are given the function $f(x) = 5^{2x}$ and we need to evaluate it at $x = -1, 0, 2$. This involves substituting these values into the function and simplifying.
2. Calculating f(-1)
First, let's evaluate $f(-1)$. We have $f(-1) = 5^{2(-1)} = 5^{-2}$. Since $5^{-2} = \frac{1}{5^2} = \frac{1}{25}$, we get $f(-1) = \frac{1}{25}$.
3. Calculating f(0)
Next, let's evaluate $f(0)$. We have $f(0) = 5^{2(0)} = 5^{0}$. Any non-zero number raised to the power of 0 is 1, so $f(0) = 1$.
4. Calculating f(2)
Finally, let's evaluate $f(2)$. We have $f(2) = 5^{2(2)} = 5^{4}$. Since $5^4 = 5 \times 5 \times 5 \times 5 = 625$, we get $f(2) = 625$.
5. Final Answer
Therefore, $f(-1) = \frac{1}{25}$, $f(0) = 1$, and $f(2) = 625$.
### Examples
Exponential functions are used to model various real-world phenomena, such as population growth, radioactive decay, and compound interest. For example, if a bacteria population doubles every hour, the population size can be modeled by an exponential function. Similarly, the amount of money in a bank account with compound interest grows exponentially over time.
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