What Is The Initial Value Of The Function $k(x) = 7 \cdot (1.1)^x$?A. 1.1 B. 0 C. 1 D. 7

Understanding the Function

The given function is $k(x) = 7 \cdot (1.1)^x$. This is an exponential function, where the base is 1.1 and the coefficient is 7. The function represents a relationship between the input variable x and the output value k(x).

Identifying the Initial Value

To find the initial value of the function, we need to determine the value of k(x) when x is equal to 0. This is because the initial value of a function is the value of the function at its starting point, which is typically x = 0.

Calculating the Initial Value

To calculate the initial value of the function, we substitute x = 0 into the function:

$$k(0) = 7 \cdot (1.1)^0$$

Simplifying the Expression

Since any number raised to the power of 0 is equal to 1, we can simplify the expression:

$$k(0) = 7 \cdot 1$$

Evaluating the Expression

Now, we can evaluate the expression to find the initial value of the function:

$$k(0) = 7$$

Conclusion

The initial value of the function $k(x) = 7 \cdot (1.1)^x$ is 7. This is the value of the function when x is equal to 0.

Comparison with Options

Let's compare the calculated initial value with the given options:

• A. 1.1: This is the base of the exponential function, not the initial value.
• B. 0: This is not the initial value, as the function is not equal to 0 when x is equal to 0.
• C. 1: This is not the initial value, as the function is equal to 7 when x is equal to 0, not 1.
• D. 7: This is the correct initial value of the function.

Final Answer

The final answer is D. 7.

Frequently Asked Questions

Q1: What is the initial value of the function $k(x) = 7 \cdot (1.1)^x$?

A1: The initial value of the function is 7. This is the value of the function when x is equal to 0.

Q2: How do you calculate the initial value of an exponential function?

A2: To calculate the initial value of an exponential function, you substitute x = 0 into the function and simplify the expression.

Q3: What is the base of the exponential function $k(x) = 7 \cdot (1.1)^x$?

A3: The base of the exponential function is 1.1.

Q4: Is the initial value of the function equal to the base?

A4: No, the initial value of the function is not equal to the base. The initial value is the value of the function when x is equal to 0, while the base is the constant that is raised to the power of x.

Q5: Can you provide an example of how to calculate the initial value of a function?

A5: Yes, let's consider the function $f(x) = 3 \cdot (2)^x$. To calculate the initial value, we substitute x = 0 into the function:

$$f(0) = 3 \cdot (2)^0$$

Since any number raised to the power of 0 is equal to 1, we can simplify the expression:

$$f(0) = 3 \cdot 1$$

$$f(0) = 3$$

Therefore, the initial value of the function is 3.

Q6: What is the significance of the initial value of a function?

A6: The initial value of a function is the value of the function at its starting point, which is typically x = 0. It provides a reference point for understanding the behavior of the function.

Q7: Can you provide a real-world example of how the initial value of a function is used?

A7: Yes, let's consider a population growth model. The initial value of the function represents the initial population size, which is used as a reference point to understand the growth rate of the population over time.

Q8: How do you determine the initial value of a function in a real-world scenario?

A8: To determine the initial value of a function in a real-world scenario, you need to identify the starting point of the function, which is typically x = 0. You then substitute x = 0 into the function and simplify the expression to find the initial value.

Q9: Can you provide a mathematical proof of the initial value of the function $k(x) = 7 \cdot (1.1)^x$?

A9: Yes, let's consider the function $k(x) = 7 \cdot (1.1)^x$. To prove that the initial value is 7, we can use the following mathematical proof:

$$k(0) = 7 \cdot (1.1)^0$$

Since any number raised to the power of 0 is equal to 1, we can simplify the expression:

$$k(0) = 7 \cdot 1$$

$$k(0) = 7$$

Therefore, the initial value of the function is 7.

Q10: What is the final answer to the initial value of the function $k(x) = 7 \cdot (1.1)^x$?

A10: The final answer is 7.