Evaluating the Function for Each Input

Oct 13, 2025 by miodeya.com 140 views

Introduction

In mathematics, functions are used to describe the relationship between variables. A function is a rule that assigns to each input a unique output. In this article, we will evaluate the function $f(x) = 2^x$ for each input listed in the table. This function is an exponential function, where the base is 2 and the exponent is x.

The Function $f(x) = 2^x$

The function $f(x) = 2^x$ is an exponential function, where the base is 2 and the exponent is x. This function can be evaluated for any real number x. The function is defined as:

f(x) = 2^x

Evaluating the Function for Each Input

The table lists the input values for x and the corresponding output values for f(x). We will evaluate the function for each input listed in the table.

Input x = 0

For x = 0, we have:

f(0) = 2^0

Since any number raised to the power of 0 is 1, we have:

f(0) = 1

Input x = 1

For x = 1, we have:

f(1) = 2^1

Since any number raised to the power of 1 is itself, we have:

f(1) = 2

Input x = 2

For x = 2, we have:

f(2) = 2^2

Since any number raised to the power of 2 is the square of the number, we have:

f(2) = 4

Input x = 3

For x = 3, we have:

f(3) = 2^3

Since any number raised to the power of 3 is the cube of the number, we have:

f(3) = 8

Input x = 4

For x = 4, we have:

f(4) = 2^4

Since any number raised to the power of 4 is the fourth power of the number, we have:

f(4) = 16

Conclusion

In this article, we evaluated the function $f(x) = 2^x$ for each input listed in the table. We found that the function can be evaluated for any real number x, and the output values are 1, 2, 4, 8, and 16 for the input values 0, 1, 2, 3, and 4, respectively.

Table of Results

x	f(x)
0	1
1	2
2	4
3	8
4	16

Discussion

Mathematical Properties

The function $f(x) = 2^x$ has several mathematical properties. Some of these properties include:

One-to-one correspondence: The function is one-to-one, meaning that each input value corresponds to a unique output value.
Monotonicity: The function is monotonic, meaning that it is either increasing or decreasing.
Continuity: The function is continuous, meaning that it can be evaluated for any real number x.

Real-World Applications

The function $f(x) = 2^x$ has several real-world applications. Some of these applications include:

Finance: The function can be used to model the growth of an investment over time.
Science: The function can be used to model the growth of a population over time.
Engineering: The function can be used to model the growth of a system over time.

Conclusion

Introduction

In our previous article, we evaluated the function $f(x) = 2^x$ for each input listed in the table. In this article, we will answer some frequently asked questions about the function.

Q: What is the domain of the function $f(x) = 2^x$ ?

A: The domain of the function $f(x) = 2^x$ is all real numbers, denoted by $(-\infty, \infty)$ .

Q: What is the range of the function $f(x) = 2^x$ ?

A: The range of the function $f(x) = 2^x$ is all positive real numbers, denoted by $(0, \infty)$ .

Q: Is the function $f(x) = 2^x$ one-to-one?

A: Yes, the function $f(x) = 2^x$ is one-to-one, meaning that each input value corresponds to a unique output value.

Q: Is the function $f(x) = 2^x$ monotonic?

A: Yes, the function $f(x) = 2^x$ is monotonic, meaning that it is either increasing or decreasing.

Q: Is the function $f(x) = 2^x$ continuous?

A: Yes, the function $f(x) = 2^x$ is continuous, meaning that it can be evaluated for any real number x.

Q: What is the value of the function $f(x) = 2^x$ at x = 0?

A: The value of the function $f(x) = 2^x$ at x = 0 is 1.

Q: What is the value of the function $f(x) = 2^x$ at x = 1?

A: The value of the function $f(x) = 2^x$ at x = 1 is 2.

Q: What is the value of the function $f(x) = 2^x$ at x = 2?

A: The value of the function $f(x) = 2^x$ at x = 2 is 4.

Q: What is the value of the function $f(x) = 2^x$ at x = 3?

A: The value of the function $f(x) = 2^x$ at x = 3 is 8.

Q: What is the value of the function $f(x) = 2^x$ at x = 4?

A: The value of the function $f(x) = 2^x$ at x = 4 is 16.

Q: Can the function $f(x) = 2^x$ be used to model the growth of a population over time?

A: Yes, the function $f(x) = 2^x$ can be used to model the growth of a population over time.

Q: Can the function $f(x) = 2^x$ be used to model the growth of an investment over time?

A: Yes, the function $f(x) = 2^x$ can be used to model the growth of an investment over time.

Conclusion

In conclusion, the function $f(x) = 2^x$ is an exponential function, where the base is 2 and the exponent is x. This function can be evaluated for any real number x, and the output values are 1, 2, 4, 8, and 16 for the input values 0, 1, 2, 3, and 4, respectively. The function has several mathematical properties, including one-to-one correspondence, monotonicity, and continuity. The function also has several real-world applications, including finance, science, and engineering.

Table of Results

x	f(x)
0	1
1	2
2	4
3	8
4	16

Discussion

The function $f(x) = 2^x$ is an exponential function, where the base is 2 and the exponent is x. This function can be evaluated for any real number x. The output values are 1, 2, 4, 8, and 16 for the input values 0, 1, 2, 3, and 4, respectively. The function has several mathematical properties, including one-to-one correspondence, monotonicity, and continuity. The function also has several real-world applications, including finance, science, and engineering.