A Table Representing The Function $f(x) = 2\left(\frac{3}{2}\right)^x$ Is Shown Below.$\[ \begin{tabular}{|c|c|} \hline $x$ & $f(x)$ \\ \hline 0 & 2 \\ \hline 1 & 3 \\ \hline 2 & 4.5 \\ \hline 3 & 6.75

Introduction

In mathematics, functions are used to describe the relationship between variables. A function can be represented in various ways, including algebraically, graphically, and tabularly. In this article, we will focus on a table representing the function $f(x) = 2\left(\frac{3}{2}\right)^x$. This function is an example of an exponential function, which is a function that has the form $f(x) = a^x$, where $a$ is a positive constant.

Understanding the Function

The function $f(x) = 2\left(\frac{3}{2}\right)^x$ can be broken down into two parts: the base $\frac{3}{2}$ and the exponent $x$. The base $\frac{3}{2}$ is a positive constant, and the exponent $x$ is the variable. When we raise the base to the power of the exponent, we get the value of the function.

The Table

The table below represents the function $f(x) = 2\left(\frac{3}{2}\right)^x$ for different values of $x$.

Analyzing the Table

From the table, we can see that as $x$ increases, the value of $f(x)$ also increases. This is because the base $\frac{3}{2}$ is greater than 1, and when we raise it to a positive exponent, the result is greater than 1.

Calculating the Values

To calculate the values of $f(x)$, we can use the formula $f(x) = 2\left(\frac{3}{2}\right)^x$. For example, to calculate $f(1)$, we can plug in $x = 1$ into the formula:

$f(1) = 2\left(\frac{3}{2}\right)^1$ $f(1) = 2 \cdot \frac{3}{2}$ $f(1) = 3$

Graphing the Function

The graph of the function $f(x) = 2\left(\frac{3}{2}\right)^x$ is an exponential curve that passes through the points (0, 2), (1, 3), (2, 4.5), and (3, 6.75). The graph can be used to visualize the behavior of the function and to make predictions about its values for different inputs.

Properties of the Function

The function $f(x) = 2\left(\frac{3}{2}\right)^x$ has several properties that make it useful in mathematics and other fields. Some of these properties include:

• Domain: The domain of the function is all real numbers, which means that the function can take on any real value as input.
• Range: The range of the function is all positive real numbers, which means that the function can only take on positive values as output.
• One-to-one: The function is one-to-one, which means that each input corresponds to a unique output.
• Continuous: The function is continuous, which means that it can be drawn without lifting the pencil from the paper.

Applications of the Function

The function $f(x) = 2\left(\frac{3}{2}\right)^x$ has several applications in mathematics and other fields. Some of these applications include:

• Modeling population growth: The function can be used to model the growth of a population over time.
• Modeling chemical reactions: The function can be used to model the rate of a chemical reaction over time.
• Modeling financial growth: The function can be used to model the growth of an investment over time.

Conclusion

Q&A

Q: What is the domain of the function $f(x) = 2\left(\frac{3}{2}\right)^x$? A: The domain of the function is all real numbers, which means that the function can take on any real value as input.

Q: What is the range of the function $f(x) = 2\left(\frac{3}{2}\right)^x$? A: The range of the function is all positive real numbers, which means that the function can only take on positive values as output.

Q: Is the function $f(x) = 2\left(\frac{3}{2}\right)^x$ one-to-one? A: Yes, the function is one-to-one, which means that each input corresponds to a unique output.

Q: Is the function $f(x) = 2\left(\frac{3}{2}\right)^x$ continuous? A: Yes, the function is continuous, which means that it can be drawn without lifting the pencil from the paper.

Q: How can the function $f(x) = 2\left(\frac{3}{2}\right)^x$ be used to model population growth? A: The function can be used to model the growth of a population over time by using the formula $f(x) = 2\left(\frac{3}{2}\right)^x$ to represent the population size at time $x$.

Q: How can the function $f(x) = 2\left(\frac{3}{2}\right)^x$ be used to model chemical reactions? A: The function can be used to model the rate of a chemical reaction over time by using the formula $f(x) = 2\left(\frac{3}{2}\right)^x$ to represent the rate of reaction at time $x$.

Q: How can the function $f(x) = 2\left(\frac{3}{2}\right)^x$ be used to model financial growth? A: The function can be used to model the growth of an investment over time by using the formula $f(x) = 2\left(\frac{3}{2}\right)^x$ to represent the investment value at time $x$.

Q: What is the value of $f(0)$? A: The value of $f(0)$ is 2, which can be calculated using the formula $f(x) = 2\left(\frac{3}{2}\right)^x$ by plugging in $x = 0$.

Q: What is the value of $f(1)$? A: The value of $f(1)$ is 3, which can be calculated using the formula $f(x) = 2\left(\frac{3}{2}\right)^x$ by plugging in $x = 1$.

Q: What is the value of $f(2)$? A: The value of $f(2)$ is 4.5, which can be calculated using the formula $f(x) = 2\left(\frac{3}{2}\right)^x$ by plugging in $x = 2$.

Q: What is the value of $f(3)$? A: The value of $f(3)$ is 6.75, which can be calculated using the formula $f(x) = 2\left(\frac{3}{2}\right)^x$ by plugging in $x = 3$.

Conclusion

In conclusion, the function $f(x) = 2\left(\frac{3}{2}\right)^x$ is an example of an exponential function that can be represented in various ways, including algebraically, graphically, and tabularly. The function has several properties that make it useful in mathematics and other fields, and it has several applications in modeling population growth, chemical reactions, and financial growth.