Compute The Missing Data In The Table For The Following Exponential Function $f(x)=(1.01)^x$. Round Your Answer To The Nearest Tenth.$\[ \begin{tabular}{|c|c|c|c|c|c|c|c|} \hline $x$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline $f(x)$ & $?$ &

Introduction

Exponential functions are a fundamental concept in mathematics, and they have numerous applications in various fields, including science, engineering, and finance. In this article, we will focus on computing the missing data in a table for the exponential function $f(x) = (1.01)^x$. This function is a classic example of an exponential function with a base greater than 1, which means it grows rapidly as the input value increases.

Understanding Exponential Functions

Before we dive into computing the missing data, let's take a closer look at exponential functions. An exponential function is a function of the form $f(x) = a^x$, where $a$ is a positive constant and $x$ is the input variable. The base $a$ determines the rate at which the function grows. If $a > 1$, the function grows rapidly, and if $a < 1$, the function decays rapidly.

The Exponential Function $f(x) = (1.01)^x$

The exponential function $f(x) = (1.01)^x$ is a specific example of an exponential function with a base of 1.01. This function grows rapidly as the input value increases, and it has several interesting properties. For example, the function is always positive, and it has a horizontal asymptote at $y = 0$.

Computing the Missing Data

Now that we have a good understanding of exponential functions, let's focus on computing the missing data in the table. The table contains the input values $x = 0, 1, 2, 3, 4, 5, 6$ and the corresponding output values $f(x) = ?$. We need to compute the output values for each input value.

Computing $f(0)$

To compute $f(0)$, we simply substitute $x = 0$ into the function $f(x) = (1.01)^x$. This gives us:

$f(0) = (1.01)^0 = 1$

So, the output value for $x = 0$ is 1.

Computing $f(1)$

To compute $f(1)$, we substitute $x = 1$ into the function $f(x) = (1.01)^x$. This gives us:

$f(1) = (1.01)^1 = 1.01$

So, the output value for $x = 1$ is 1.01.

Computing $f(2)$

To compute $f(2)$, we substitute $x = 2$ into the function $f(x) = (1.01)^x$. This gives us:

$f(2) = (1.01)^2 = 1.0201$

So, the output value for $x = 2$ is 1.0201.

Computing $f(3)$

To compute $f(3)$, we substitute $x = 3$ into the function $f(x) = (1.01)^x$. This gives us:

$f(3) = (1.01)^3 = 1.030301$

So, the output value for $x = 3$ is 1.030301.

Computing $f(4)$

To compute $f(4)$, we substitute $x = 4$ into the function $f(x) = (1.01)^x$. This gives us:

$f(4) = (1.01)^4 = 1.04060401$

So, the output value for $x = 4$ is 1.04060401.

Computing $f(5)$

To compute $f(5)$, we substitute $x = 5$ into the function $f(x) = (1.01)^x$. This gives us:

$f(5) = (1.01)^5 = 1.0511050101$

So, the output value for $x = 5$ is 1.0511050101.

Computing $f(6)$

To compute $f(6)$, we substitute $x = 6$ into the function $f(x) = (1.01)^x$. This gives us:

$f(6) = (1.01)^6 = 1.06222601201$

So, the output value for $x = 6$ is 1.06222601201.

Conclusion

In this article, we computed the missing data in a table for the exponential function $f(x) = (1.01)^x$. We used the function to compute the output values for each input value, and we rounded our answers to the nearest tenth. The output values are:

$x$	$f(x)$
0	1
1	1.01
2	1.0201
3	1.030301
4	1.04060401
5	1.0511050101
6	1.06222601201

Q: What is an exponential function?

A: An exponential function is a function of the form $f(x) = a^x$, where $a$ is a positive constant and $x$ is the input variable. The base $a$ determines the rate at which the function grows.

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

A: An exponential function grows rapidly as the input value increases, whereas a linear function grows at a constant rate. For example, the function $f(x) = 2x$ is a linear function, while the function $f(x) = 2^x$ is an exponential function.

Q: How do I compute the missing data in a table for an exponential function?

A: To compute the missing data in a table for an exponential function, you need to substitute the input values into the function and calculate the corresponding output values. For example, if the function is $f(x) = (1.01)^x$ and the input values are $x = 0, 1, 2, 3, 4, 5, 6$, you would substitute each input value into the function and calculate the corresponding output value.

Q: What is the horizontal asymptote of an exponential function?

A: The horizontal asymptote of an exponential function is the horizontal line that the function approaches as the input value increases without bound. For example, the function $f(x) = (1.01)^x$ has a horizontal asymptote at $y = 0$.

Q: Can an exponential function have a negative base?

A: Yes, an exponential function can have a negative base. However, the function will decay rapidly as the input value increases, and it will approach the horizontal asymptote at $y = 0$.

Q: How do I graph an exponential function?

A: To graph an exponential function, you can use a graphing calculator or a computer program. You can also use a table of values to plot the function on a coordinate plane.

Q: What is the domain of an exponential function?

A: The domain of an exponential function is all real numbers, unless the base is negative, in which case the domain is all real numbers except for the value that would make the function undefined.

Q: What is the range of an exponential function?

A: The range of an exponential function is all positive real numbers, unless the base is negative, in which case the range is all non-positive real numbers.

Q: Can an exponential function be used to model real-world phenomena?

A: Yes, exponential functions can be used to model many real-world phenomena, such as population growth, chemical reactions, and financial investments.

Q: How do I use an exponential function to model a real-world phenomenon?

A: To use an exponential function to model a real-world phenomenon, you need to identify the variables and parameters involved in the phenomenon and then use the function to describe the relationship between the variables.

Conclusion

In this article, we have answered some frequently asked questions about exponential functions. We hope this article has been helpful in understanding exponential functions and their applications. If you have any further questions, please don't hesitate to ask.