Answer :
- Substitute $x=3$ into the function $f(x)=\left(\frac{1}{7}\right)\left(7^x\right)$.
- Calculate $7^3 = 343$.
- Simplify the expression $\left(\frac{1}{7}\right)(343)$.
- The final answer is $\boxed{49}$.
### Explanation
1. Understanding the problem
We are given the function $f(x)=\left(\frac{1}{7}\right)\left(7^x\right)$ and asked to find the value of $f(3)$. This involves substituting $x=3$ into the function and simplifying the expression.
2. Substituting x=3
To find $f(3)$, we substitute $x=3$ into the function: $$f(3) = \left(\frac{1}{7}\right)\left(7^3\right)$$.
3. Calculating 7^3
Now, we calculate $7^3$, which is $7 \times 7 \times 7 = 343$. So, we have: $$f(3) = \left(\frac{1}{7}\right)(343)$$.
4. Simplifying the expression
Next, we simplify the expression by dividing 343 by 7: $$\frac{343}{7} = 49$$. Therefore, $f(3) = 49$.
5. Final Answer
Comparing our result with the given options, we find that $f(3) = 49$ corresponds to option B.
### Examples
Exponential functions are used to model various real-world phenomena, such as population growth, radioactive decay, and compound interest. For example, if a population of bacteria 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. Understanding how to evaluate exponential functions is crucial for making predictions and analyzing these phenomena.
- Calculate $7^3 = 343$.
- Simplify the expression $\left(\frac{1}{7}\right)(343)$.
- The final answer is $\boxed{49}$.
### Explanation
1. Understanding the problem
We are given the function $f(x)=\left(\frac{1}{7}\right)\left(7^x\right)$ and asked to find the value of $f(3)$. This involves substituting $x=3$ into the function and simplifying the expression.
2. Substituting x=3
To find $f(3)$, we substitute $x=3$ into the function: $$f(3) = \left(\frac{1}{7}\right)\left(7^3\right)$$.
3. Calculating 7^3
Now, we calculate $7^3$, which is $7 \times 7 \times 7 = 343$. So, we have: $$f(3) = \left(\frac{1}{7}\right)(343)$$.
4. Simplifying the expression
Next, we simplify the expression by dividing 343 by 7: $$\frac{343}{7} = 49$$. Therefore, $f(3) = 49$.
5. Final Answer
Comparing our result with the given options, we find that $f(3) = 49$ corresponds to option B.
### Examples
Exponential functions are used to model various real-world phenomena, such as population growth, radioactive decay, and compound interest. For example, if a population of bacteria 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. Understanding how to evaluate exponential functions is crucial for making predictions and analyzing these phenomena.
