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Answer :
- Substitute $x=3$ into the function $f(x)=\left(\frac{1}{9}\right)\left(9^x\right)$.
- Calculate $9^3 = 729$.
- Simplify the expression $\left(\frac{1}{9}\right)(729)$.
- The final answer is $\boxed{81}$.
### Explanation
1. Understanding the problem
We are given the function $f(x)=\left(\frac{1}{9}\right)\left(9^x\right)$ and we need to find the value of $f(3)$. This means we need to substitute $x=3$ into the function and simplify the expression.
2. Substitution
Substitute $x=3$ into the function:$$f(3) = \left(\frac{1}{9}\right)\left(9^3\right)$$.
3. Calculating 9^3
Now, we need to calculate $9^3$. We know that $9^3 = 9 \times 9 \times 9 = 81 \times 9 = 729$. So, we have:$$f(3) = \left(\frac{1}{9}\right)(729)$$.
4. Simplifying the expression
Next, we divide 729 by 9:$$\frac{729}{9} = 81$$.Therefore, $f(3) = 81$.
5. Final Answer
The value of $f(3)$ is 81. Comparing this with the given options, we see that option B is the correct answer.
### Examples
Imagine you are tracking the growth of a plant where its height each day is given by the function f(x)=(1/9)(9^x). Evaluating f(3) tells you the height of the plant on the third day. This type of exponential function can model growth in various scenarios, such as population growth or compound interest, where understanding the value at a specific point in time is crucial for making predictions and decisions.
- Calculate $9^3 = 729$.
- Simplify the expression $\left(\frac{1}{9}\right)(729)$.
- The final answer is $\boxed{81}$.
### Explanation
1. Understanding the problem
We are given the function $f(x)=\left(\frac{1}{9}\right)\left(9^x\right)$ and we need to find the value of $f(3)$. This means we need to substitute $x=3$ into the function and simplify the expression.
2. Substitution
Substitute $x=3$ into the function:$$f(3) = \left(\frac{1}{9}\right)\left(9^3\right)$$.
3. Calculating 9^3
Now, we need to calculate $9^3$. We know that $9^3 = 9 \times 9 \times 9 = 81 \times 9 = 729$. So, we have:$$f(3) = \left(\frac{1}{9}\right)(729)$$.
4. Simplifying the expression
Next, we divide 729 by 9:$$\frac{729}{9} = 81$$.Therefore, $f(3) = 81$.
5. Final Answer
The value of $f(3)$ is 81. Comparing this with the given options, we see that option B is the correct answer.
### Examples
Imagine you are tracking the growth of a plant where its height each day is given by the function f(x)=(1/9)(9^x). Evaluating f(3) tells you the height of the plant on the third day. This type of exponential function can model growth in various scenarios, such as population growth or compound interest, where understanding the value at a specific point in time is crucial for making predictions and decisions.
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