Completing the Table of Values

Oct 13, 2025 by recoveryshake.com 66 views

Exploring the Functions f(x) and g(x)

In mathematics, functions are a crucial concept that helps us understand the relationship between variables. In this article, we will delve into two specific functions, f(x) and g(x), and explore their properties and behavior.

Function f(x) = 2x

The function f(x) = 2x is a linear function that takes an input value x and multiplies it by 2. This function is characterized by its constant rate of change, which is 2. In other words, for every unit increase in x, the output value of f(x) increases by 2.

Function g(x) = 2^x

The function g(x) = 2^x is an exponential function that takes an input value x and raises 2 to the power of x. This function is characterized by its rapid growth rate, which increases exponentially as x increases.

Completing the Table of Values

To gain a better understanding of these functions, we need to complete the table of values for each function.

x	f(x) = 2x	g(x) = 2^x
0	0	1
1	2	2
2	4	4
3	6	8
4	8	16
5	10	32
6	12	64
7	14	128
8	16	256
9	18	512

As we can see from the table, the function f(x) = 2x is a linear function that increases at a constant rate of 2. On the other hand, the function g(x) = 2^x is an exponential function that grows rapidly as x increases.

Discussion

The functions f(x) and g(x) have some interesting properties that are worth exploring. For example, the function f(x) = 2x is a one-to-one function, meaning that each output value corresponds to a unique input value. In contrast, the function g(x) = 2^x is not one-to-one, as multiple input values can produce the same output value.

Another interesting property of the function g(x) = 2^x is that it is an increasing function, meaning that as x increases, g(x) also increases. This is in contrast to the function f(x) = 2x, which is also increasing, but at a constant rate.

Graphing the Functions

To visualize the behavior of these functions, we can graph them on a coordinate plane.

Graph of f(x) = 2x

The graph of f(x) = 2x is a straight line with a slope of 2. The line passes through the origin (0, 0) and has a y-intercept of 0.

Graph of g(x) = 2^x

The graph of g(x) = 2^x is an exponential curve that grows rapidly as x increases. The curve passes through the point (0, 1) and has a y-intercept of 1.

Conclusion

In conclusion, the functions f(x) = 2x and g(x) = 2^x are two distinct functions with different properties and behaviors. The function f(x) = 2x is a linear function that increases at a constant rate, while the function g(x) = 2^x is an exponential function that grows rapidly as x increases. By exploring these functions and their properties, we can gain a deeper understanding of the mathematical concepts that underlie them.

Applications of the Functions

The functions f(x) = 2x and g(x) = 2^x have numerous applications in mathematics and other fields. For example, the function f(x) = 2x is used in finance to calculate interest rates, while the function g(x) = 2^x is used in computer science to model population growth.

Real-World Examples

Here are some real-world examples of how the functions f(x) = 2x and g(x) = 2^x are used:

Finance: The function f(x) = 2x is used to calculate interest rates on loans and investments. For example, if you deposit $100 into a savings account with a 2% interest rate, the function f(x) = 2x would calculate the interest earned as $2.
Computer Science: The function g(x) = 2^x is used to model population growth in computer simulations. For example, if a population grows at a rate of 2% per year, the function g(x) = 2^x would calculate the population size after 10 years as 2^10 = 1024.
Biology: The function g(x) = 2^x is used to model the growth of bacteria in a petri dish. For example, if a culture of bacteria grows at a rate of 2% per hour, the function g(x) = 2^x would calculate the number of bacteria after 10 hours as 2^10 = 1024.

Conclusion

In conclusion, the functions f(x) = 2x and g(x) = 2^x are two distinct functions with different properties and behaviors. By exploring these functions and their applications, we can gain a deeper understanding of the mathematical concepts that underlie them.