Answer :
- Substitute $x = \frac{1}{2}$ into the function $f(x) = 3 \cdot 25^x$.
- Simplify the exponent: $25^{\frac{1}{2}} = \sqrt{25} = 5$.
- Multiply the result by 3: $3 \cdot 5 = 15$.
- The value of $f\left(\frac{1}{2}\right)$ is $\boxed{15}$.
### Explanation
1. Understanding the problem
We are given the exponential function $f(x) = 3 \cdot 25^x$ and we want to find the value of $f\left(\frac{1}{2}\right)$. This means we need to substitute $\frac{1}{2}$ for $x$ in the function.
2. Substitution
Substitute $x = \frac{1}{2}$ into the function: $$f\left(\frac{1}{2}\right) = 3 \cdot 25^{\frac{1}{2}}$$.
3. Simplifying the exponent
We know that $25^{\frac{1}{2}}$ is the same as the square root of 25, which is $\sqrt{25}$. The square root of 25 is 5, since $5 \cdot 5 = 25$. So, we have: $$f\left(\frac{1}{2}\right) = 3 \cdot 5$$.
4. Final Calculation
Now, we just need to multiply 3 by 5: $$3 \cdot 5 = 15$$. Therefore, $f\left(\frac{1}{2}\right) = 15$.
5. Conclusion
The value of the function $f(x) = 3 \cdot 25^x$ at $x = \frac{1}{2}$ is 15.
### Examples
Exponential functions are used to model various real-world phenomena, such as population growth, radioactive decay, and compound interest. For example, if you invest money in an account that earns compound interest, the amount of money you have after a certain period of time can be modeled by an exponential function. Understanding how to evaluate exponential functions is essential for making informed financial decisions.
- Simplify the exponent: $25^{\frac{1}{2}} = \sqrt{25} = 5$.
- Multiply the result by 3: $3 \cdot 5 = 15$.
- The value of $f\left(\frac{1}{2}\right)$ is $\boxed{15}$.
### Explanation
1. Understanding the problem
We are given the exponential function $f(x) = 3 \cdot 25^x$ and we want to find the value of $f\left(\frac{1}{2}\right)$. This means we need to substitute $\frac{1}{2}$ for $x$ in the function.
2. Substitution
Substitute $x = \frac{1}{2}$ into the function: $$f\left(\frac{1}{2}\right) = 3 \cdot 25^{\frac{1}{2}}$$.
3. Simplifying the exponent
We know that $25^{\frac{1}{2}}$ is the same as the square root of 25, which is $\sqrt{25}$. The square root of 25 is 5, since $5 \cdot 5 = 25$. So, we have: $$f\left(\frac{1}{2}\right) = 3 \cdot 5$$.
4. Final Calculation
Now, we just need to multiply 3 by 5: $$3 \cdot 5 = 15$$. Therefore, $f\left(\frac{1}{2}\right) = 15$.
5. Conclusion
The value of the function $f(x) = 3 \cdot 25^x$ at $x = \frac{1}{2}$ is 15.
### Examples
Exponential functions are used to model various real-world phenomena, such as population growth, radioactive decay, and compound interest. For example, if you invest money in an account that earns compound interest, the amount of money you have after a certain period of time can be modeled by an exponential function. Understanding how to evaluate exponential functions is essential for making informed financial decisions.
