Answer :
- Substitute $x=3$ into the function $f(x) = \left(\frac{1}{7}\right)(7^x)$.
- Calculate $7^3 = 343$.
- Multiply $\frac{1}{7}$ by $343$ to get $\frac{343}{7}$.
- Simplify the fraction to find $f(3) = \boxed{49}$.
### Explanation
1. Understanding the problem
We are given the function $f(x) = \left(\frac{1}{7}\right)(7^x)$ and we need to find the value of $f(3)$. This means we need to substitute $x=3$ into the function and simplify.
2. Substitution
Substitute $x=3$ into the function: $f(3) = \left(\frac{1}{7}\right)(7^3)$.
3. Calculating 7 cubed
Calculate $7^3$: $7^3 = 7 \times 7 \times 7 = 49 \times 7 = 343$. So, $f(3) = \left(\frac{1}{7}\right)(343)$.
4. Simplifying the expression
Now, multiply $\frac{1}{7}$ by $343$: $f(3) = \frac{343}{7}$. To simplify this fraction, we can divide 343 by 7. The result is 49.
5. Final Answer
Therefore, $f(3) = 49$.
### Examples
Imagine you are tracking the growth of a plant where its height each day is given by the function $f(x) = \frac{1}{7} \cdot 7^x$, where $x$ is the number of days. Finding $f(3)$ tells you the height of the plant on the third day. This type of exponential function can model growth in various real-life scenarios, such as population growth or compound interest.
- Calculate $7^3 = 343$.
- Multiply $\frac{1}{7}$ by $343$ to get $\frac{343}{7}$.
- Simplify the fraction to find $f(3) = \boxed{49}$.
### Explanation
1. Understanding the problem
We are given the function $f(x) = \left(\frac{1}{7}\right)(7^x)$ and we need to find the value of $f(3)$. This means we need to substitute $x=3$ into the function and simplify.
2. Substitution
Substitute $x=3$ into the function: $f(3) = \left(\frac{1}{7}\right)(7^3)$.
3. Calculating 7 cubed
Calculate $7^3$: $7^3 = 7 \times 7 \times 7 = 49 \times 7 = 343$. So, $f(3) = \left(\frac{1}{7}\right)(343)$.
4. Simplifying the expression
Now, multiply $\frac{1}{7}$ by $343$: $f(3) = \frac{343}{7}$. To simplify this fraction, we can divide 343 by 7. The result is 49.
5. Final Answer
Therefore, $f(3) = 49$.
### Examples
Imagine you are tracking the growth of a plant where its height each day is given by the function $f(x) = \frac{1}{7} \cdot 7^x$, where $x$ is the number of days. Finding $f(3)$ tells you the height of the plant on the third day. This type of exponential function can model growth in various real-life scenarios, such as population growth or compound interest.
