Select The Correct Answer.Exponential Function $f$ Is Represented By The Table:$\[ \begin{array}{|c|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 & 4 \\ \hline f(x) & 15 & 7 & 3 & 1 & 0 \\ \hline \end{array} \\]Function $g$ Is An

Introduction to Exponential Functions

Exponential functions are a fundamental concept in mathematics, used to describe the growth or decay of a quantity over time. They are represented by the equation $f(x) = ab^x$, where $a$ is the initial value, $b$ is the base, and $x$ is the exponent. In this article, we will explore the concept of exponential functions and their representations, using a table to illustrate the relationship between the input and output values.

Representing Exponential Functions Using Tables

A table can be used to represent an exponential function, where the input values are listed in the first column and the corresponding output values are listed in the second column. For example, the table below represents the exponential function $f(x) = 15 \cdot 2^x$.

x	f(x)
0	15
1	30
2	60
3	120
4	240

Analyzing the Table

From the table, we can see that the output value $f(x)$ increases exponentially as the input value $x$ increases. This is because the base $b$ is greater than 1, which causes the function to grow rapidly. We can also observe that the output value is always positive, which is a characteristic of exponential functions.

Finding the Correct Representation

Now, let's consider the table provided in the problem statement.

x	f(x)
0	15
1	7
2	3
3	1
4	0

We need to find the correct representation of the exponential function $g$. To do this, we can analyze the table and look for patterns or relationships between the input and output values.

Identifying the Pattern

Upon closer inspection, we can see that the output value $f(x)$ decreases exponentially as the input value $x$ increases. This is because the base $b$ is less than 1, which causes the function to decay rapidly. We can also observe that the output value is always positive, which is a characteristic of exponential functions.

Determining the Correct Representation

Based on the analysis, we can conclude that the correct representation of the exponential function $g$ is $g(x) = 15 \cdot \left(\frac{1}{2}\right)^x$. This function has a base of $\frac{1}{2}$, which causes the output value to decrease exponentially as the input value increases.

Verifying the Representation

To verify the representation, we can plug in the input values from the table and calculate the corresponding output values using the function $g(x) = 15 \cdot \left(\frac{1}{2}\right)^x$.

x	g(x)
0	15
1	7.5
2	3.75
3	1.875
4	0.9375

The calculated output values match the values listed in the table, which confirms that the correct representation of the exponential function $g$ is indeed $g(x) = 15 \cdot \left(\frac{1}{2}\right)^x$.

Conclusion

In conclusion, we have analyzed the table representing the exponential function $g$ and determined the correct representation of the function. We have also verified the representation by plugging in the input values and calculating the corresponding output values. This exercise has helped us understand the concept of exponential functions and their representations, and has provided a practical example of how to analyze and represent exponential functions using tables.

Key Takeaways

• Exponential functions can be represented using tables, where the input values are listed in the first column and the corresponding output values are listed in the second column.
• The base of an exponential function determines whether the function grows or decays.
• The correct representation of an exponential function can be determined by analyzing the table and looking for patterns or relationships between the input and output values.
• Verifying the representation by plugging in the input values and calculating the corresponding output values can confirm the correctness of the representation.

Further Reading

For further reading on exponential functions and their representations, we recommend the following resources:

• Khan Academy: Exponential Functions
• Math Is Fun: Exponential Functions
• Wolfram MathWorld: Exponential Function

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about exponential functions and their representations.

Q: What is an exponential function?

A: An exponential function is a mathematical function that describes the growth or decay of a quantity over time. It is represented by the equation $f(x) = ab^x$, where $a$ is the initial value, $b$ is the base, and $x$ is the exponent.

Q: What is the base of an exponential function?

A: The base of an exponential function is the value that is raised to the power of the exponent. For example, in the function $f(x) = 2^x$, the base is 2.

Q: What is the exponent of an exponential function?

A: The exponent of an exponential function is the value that is multiplied by the base. For example, in the function $f(x) = 2^x$, the exponent is $x$.

Q: What is the initial value of an exponential function?

A: The initial value of an exponential function is the value of the function when the exponent is 0. For example, in the function $f(x) = 2^x$, the initial value is 1.

Q: How do I determine the correct representation of an exponential function?

A: To determine the correct representation of an exponential function, you need to analyze the table and look for patterns or relationships between the input and output values. You can also use the formula $f(x) = ab^x$ to determine the correct representation.

Q: How do I verify the representation of an exponential function?

A: To verify the representation of an exponential function, you need to plug in the input values from the table and calculate the corresponding output values using the function. If the calculated output values match the values listed in the table, then the representation is correct.

Q: What is the difference between an exponential function and a linear function?

A: An exponential function is a function that describes the growth or decay of a quantity over time, while a linear function is a function that describes a straight line. Exponential functions have a base that is raised to the power of the exponent, while linear functions have a slope that is multiplied by the input value.

Q: Can an exponential function have a negative base?

A: Yes, an exponential function can have a negative base. For example, the function $f(x) = (-2)^x$ has a negative base of -2.

Q: Can an exponential function have a fractional base?

A: Yes, an exponential function can have a fractional base. For example, the function $f(x) = \left(\frac{1}{2}\right)^x$ has a fractional base of $\frac{1}{2}$.

Q: Can an exponential function have a zero base?

A: No, an exponential function cannot have a zero base. This is because the base of an exponential function must be a non-zero value.

Conclusion

In conclusion, we have answered some of the most frequently asked questions about exponential functions and their representations. We hope that this article has provided you with a better understanding of exponential functions and their representations.

Key Takeaways

• Exponential functions describe the growth or decay of a quantity over time.
• The base of an exponential function determines whether the function grows or decays.
• The exponent of an exponential function determines the rate of growth or decay.
• The initial value of an exponential function is the value of the function when the exponent is 0.
• To determine the correct representation of an exponential function, you need to analyze the table and look for patterns or relationships between the input and output values.
• To verify the representation of an exponential function, you need to plug in the input values from the table and calculate the corresponding output values using the function.

Further Reading

For further reading on exponential functions and their representations, we recommend the following resources:

• Khan Academy: Exponential Functions
• Math Is Fun: Exponential Functions
• Wolfram MathWorld: Exponential Function

These resources provide a comprehensive introduction to exponential functions and their representations, and offer practical examples and exercises to help you understand the concept.