Identify The Type Of Function Represented By $f(x)=\frac{3}{8}(4)^x$.A. Quadratic B. Exponential Decay C. Linear D. Exponential Growth
In mathematics, functions are a crucial concept that helps us describe the relationship between variables. There are various types of functions, each with its unique characteristics and properties. In this article, we will delve into the world of functions and identify the type of function represented by the given equation $f(x)=\frac{3}{8}(4)^x$.
What is a Function?
A function is a relation between a set of inputs, called the domain, and a set of possible outputs, called the range. It is a rule that assigns to each input exactly one output. Functions can be represented graphically, algebraically, or verbally. In this article, we will focus on the algebraic representation of functions.
Types of Functions
Functions can be classified into several categories based on their characteristics. The main types of functions are:
- Linear Functions: These functions have a constant rate of change and can be represented by the equation $f(x) = mx + b$, where is the slope and is the y-intercept.
- Quadratic Functions: These functions have a parabolic shape and can be represented by the equation $f(x) = ax^2 + bx + c$, where , , and are constants.
- Exponential Functions: These functions have a constant base and can be represented by the equation $f(x) = ab^x$, where and are constants.
- Logarithmic Functions: These functions have a constant base and can be represented by the equation $f(x) = \log_b(x)$, where is a constant.
Identifying the Type of Function
Now that we have a basic understanding of functions and their types, let's focus on the given equation $f(x)=\frac{3}{8}(4)^x$. To identify the type of function represented by this equation, we need to analyze its characteristics.
- Constant Base: The equation has a constant base of , which is a characteristic of exponential functions.
- Exponential Term: The equation contains an exponential term , which is a characteristic of exponential functions.
- No Linear or Quadratic Terms: The equation does not contain any linear or quadratic terms, which rules out linear and quadratic functions.
Conclusion
Based on the analysis of the given equation $f(x)=\frac{3}{8}(4)^x$, we can conclude that it represents an Exponential Function. The constant base of and the exponential term are the key characteristics that identify this function as an exponential function.
Why is it an Exponential Function?
The equation $f(x)=\frac{3}{8}(4)^x$ represents an exponential function because it has a constant base of and an exponential term . The constant base means that the function grows or decays at a constant rate, while the exponential term means that the function grows or decays exponentially.
What is Exponential Growth?
Exponential growth is a type of growth that occurs when a quantity increases at a constant rate. It is characterized by a rapid increase in the quantity over time. Exponential growth is often represented by the equation $f(x) = ab^x$, where and are constants.
What is Exponential Decay?
Exponential decay is a type of decay that occurs when a quantity decreases at a constant rate. It is characterized by a rapid decrease in the quantity over time. Exponential decay is often represented by the equation $f(x) = ab^x$, where and are constants.
Why is the Given Equation an Exponential Growth Function?
The given equation $f(x)=\frac{3}{8}(4)^x$ represents an exponential growth function because the base is greater than . This means that the function grows exponentially over time.
Conclusion
In conclusion, the given equation $f(x)=\frac{3}{8}(4)^x$ represents an exponential growth function. The constant base of and the exponential term are the key characteristics that identify this function as an exponential growth function.
Answer
The correct answer is:
D. Exponential growth
Final Thoughts
In our previous article, we discussed the concept of exponential functions and identified the type of function represented by the equation $f(x)=\frac{3}{8}(4)^x$. In this article, we will answer some frequently asked questions about exponential functions and growth.
Q: What is an exponential function?
A: An exponential function is a function that has a constant base and an exponential term. It is characterized by a rapid growth or decay over time.
Q: What is the difference between exponential growth and exponential decay?
A: Exponential growth occurs when a quantity increases at a constant rate, while exponential decay occurs when a quantity decreases at a constant rate.
Q: How do you identify an exponential function?
A: To identify an exponential function, look for a constant base and an exponential term. The equation should be in the form $f(x) = ab^x$, where and are constants.
Q: What is the significance of the base in an exponential function?
A: The base of an exponential function determines the rate of growth or decay. If the base is greater than 1, the function grows exponentially. If the base is less than 1, the function decays exponentially.
Q: Can you give an example of an exponential growth function?
A: Yes, the equation $f(x) = 2(3)^x$ is an example of an exponential growth function. The base 3 is greater than 1, so the function grows exponentially.
Q: Can you give an example of an exponential decay function?
A: Yes, the equation $f(x) = 2(0.5)^x$ is an example of an exponential decay function. The base 0.5 is less than 1, so the function decays exponentially.
Q: How do you graph an exponential function?
A: To graph an exponential function, use a graphing calculator or software. Plot the points on the graph and draw a smooth curve through them.
Q: What is the domain and range of an exponential function?
A: The domain of an exponential function is all real numbers, while the range is all positive real numbers.
Q: Can you give an example of a real-world application of exponential growth?
A: Yes, population growth is an example of exponential growth. When a population grows at a constant rate, it can lead to rapid growth over time.
Q: Can you give an example of a real-world application of exponential decay?
A: Yes, radioactive decay is an example of exponential decay. When a radioactive substance decays at a constant rate, it can lead to a rapid decrease in its radioactivity over time.
Q: How do you solve an exponential equation?
A: To solve an exponential equation, use the properties of exponents to isolate the variable. For example, to solve the equation $2(3)^x = 6$, divide both sides by 2 to get $(3)^x = 3$, then take the logarithm of both sides to solve for x.
Q: What is the difference between exponential growth and linear growth?
A: Exponential growth occurs when a quantity increases at a constant rate, while linear growth occurs when a quantity increases at a constant rate per unit of time.
Q: Can you give an example of a real-world application of linear growth?
A: Yes, a car's speedometer is an example of linear growth. When a car accelerates at a constant rate, its speed increases linearly over time.
Conclusion
In this article, we have answered some frequently asked questions about exponential functions and growth. We have discussed the characteristics of exponential functions, how to identify them, and how to graph them. We have also given examples of real-world applications of exponential growth and decay.