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Identifying the Type of Function Represented by $$f(x)=4 \cdot\left(\frac{1}{3}\right)^x$$

In mathematics, functions are used to describe the relationship between variables. There are various types of functions, including linear, exponential, and quadratic functions. In this article, we will focus on identifying the type of function represented by the given equation $$f(x)=4 \cdot\left(\frac{1}{3}\right)^x$$. We will analyze the characteristics of the function and determine whether it represents an increasing linear, decreasing linear, exponential decay, or exponential growth function.

Exponential functions are a type of function that can be written in the form $$f(x)=a \cdot b^x$$, where $$a$$ and $$b$$ are constants. The base $$b$$ determines the rate of growth or decay of the function. If $$b>1$$, the function represents exponential growth, while if $$0<b<1$$, the function represents exponential decay.

The given function is $$f(x)=4 \cdot\left(\frac{1}{3}\right)^x$$. To determine the type of function, we need to analyze the characteristics of the function.

Base: The base of the function is $$\frac{1}{3}$$, which is less than 1. This indicates that the function represents exponential decay.
Coefficient: The coefficient of the function is 4, which is a positive constant. This means that the function will always be positive, regardless of the value of $$x$$.
Domain: The domain of the function is all real numbers, since the base $$\frac{1}{3}$$ is a positive constant and the exponent $$x$$ can take any real value.

Based on the analysis of the function, we can conclude that the function represents an exponential decay function. This is because the base $$\frac{1}{3}$$ is less than 1, indicating that the function will decrease as $$x$$ increases.

To confirm that the function represents an exponential decay function, we can compare it with the other options.

Increasing Linear Function: An increasing linear function has a positive slope, which means that the function will increase as $$x$$ increases. However, the given function represents an exponential decay, which means that it will decrease as $$x$$ increases.
Decreasing Linear Function: A decreasing linear function has a negative slope, which means that the function will decrease as $$x$$ increases. However, the given function represents an exponential decay, which means that it will decrease at a faster rate as $$x$$ increases.
Exponential Growth Function: An exponential growth function has a base greater than 1, which means that the function will increase as $$x$$ increases. However, the given function represents an exponential decay, which means that it will decrease as $$x$$ increases.

In conclusion, the function $$f(x)=4 \cdot\left(\frac{1}{3}\right)^x$$ represents an exponential decay function. This is because the base $$\frac{1}{3}$$ is less than 1, indicating that the function will decrease as $$x$$ increases. The function will always be positive, regardless of the value of $$x$$, and the domain of the function is all real numbers.

Exponential functions can be written in the form $$f(x)=a \cdot b^x$$, where $$a$$ and $$b$$ are constants.
The base $$b$$ determines the rate of growth or decay of the function.
If $$b>1$$, the function represents exponential growth, while if $$0<b<1$$, the function represents exponential decay.
The function $$f(x)=4 \cdot\left(\frac{1}{3}\right)^x$$ represents an exponential decay function.

The final answer is C. Exponential decay.
Q&A: Identifying the Type of Function Represented by $$f(x)=4 \cdot\left(\frac{1}{3}\right)^x$$

In our previous article, we analyzed the function $$f(x)=4 \cdot\left(\frac{1}{3}\right)^x$$ and determined that it represents an exponential decay function. In this article, we will answer some frequently asked questions related to the function and provide additional insights.

Q1: What is the base of the function?

A1: The base of the function is $$\frac{1}{3}$$.

Q2: Why is the function an exponential decay function?

A2: The function is an exponential decay function because the base $$\frac{1}{3}$$ is less than 1. This means that the function will decrease as $$x$$ increases.

Q3: What is the coefficient of the function?

A3: The coefficient of the function is 4, which is a positive constant.

Q4: What is the domain of the function?

A4: The domain of the function is all real numbers, since the base $$\frac{1}{3}$$ is a positive constant and the exponent $$x$$ can take any real value.

Q5: How does the function compare to an increasing linear function?

A5: The function does not represent an increasing linear function because it decreases as $$x$$ increases, whereas an increasing linear function increases as $$x$$ increases.

Q6: How does the function compare to a decreasing linear function?

A6: The function does not represent a decreasing linear function because it decreases at a faster rate as $$x$$ increases, whereas a decreasing linear function decreases at a constant rate.

Q7: How does the function compare to an exponential growth function?

A7: The function does not represent an exponential growth function because it decreases as $$x$$ increases, whereas an exponential growth function increases as $$x$$ increases.

Q8: What is the significance of the coefficient 4 in the function?

A8: The coefficient 4 is a positive constant that ensures the function is always positive, regardless of the value of $$x$$.

Q9: Can the function be written in a different form?

A9: Yes, the function can be written in a different form using the properties of exponents. For example, $$f(x)=4 \cdot\left(\frac{1}{3}\right)^x = 4 \cdot e^{x \ln\left(\frac{1}{3}\right)}$$.

Q10: What is the relationship between the function and the natural logarithm?

A10: The function is related to the natural logarithm through the property $$e^{x \ln(b)} = b^x$$.

In conclusion, the function $$f(x)=4 \cdot\left(\frac{1}{3}\right)^x$$ represents an exponential decay function. We have answered some frequently asked questions related to the function and provided additional insights into its properties.

The base of the function is $$\frac{1}{3}$$.
The function is an exponential decay function because the base $$\frac{1}{3}$$ is less than 1.
The coefficient of the function is 4, which is a positive constant.
The domain of the function is all real numbers.
The function does not represent an increasing linear function, a decreasing linear function, or an exponential growth function.

The final answer is C. Exponential decay.