Select The Correct Answer.Which Exponential Function Does Not Have An \[$x\$\]-intercept?A. \[$f(x)=5^{x-5}-1\$\]B. \[$f(x)=5^{x-5}-5\$\]C. \[$f(x)=-5^{x-5}+5\$\]D. \[$r(x)=-5^{x-5}-1\$\]
Introduction
Exponential functions are a fundamental concept in mathematics, and understanding their properties is crucial for solving various mathematical problems. In this article, we will delve into the world of exponential functions and explore the concept of x-intercepts. We will examine four different exponential functions and determine which one does not have an x-intercept.
What are Exponential Functions?
Exponential functions are a type of mathematical function that involves an exponent, which is a number or expression raised to a power. The general form of an exponential function is:
f(x) = a^x + b
where a is the base and b is the constant term. Exponential functions can be increasing or decreasing, depending on the value of the base.
X-Intercepts: A Brief Overview
An x-intercept is a point on the graph of a function where the y-coordinate is zero. In other words, it is the point where the function crosses the x-axis. X-intercepts are an essential concept in mathematics, as they help us understand the behavior of functions and their relationship with the x-axis.
Analyzing the Exponential Functions
Let's analyze each of the four exponential functions given in the problem statement:
A. f(x) = 5^(x-5) - 1
This function has a base of 5 and a constant term of -1. To determine if it has an x-intercept, we need to find the value of x that makes the function equal to zero.
f(x) = 5^(x-5) - 1 = 0
Solving for x, we get:
5^(x-5) = 1
x - 5 = 0
x = 5
Therefore, the function f(x) = 5^(x-5) - 1 has an x-intercept at x = 5.
B. f(x) = 5^(x-5) - 5
This function also has a base of 5, but the constant term is -5. To determine if it has an x-intercept, we need to find the value of x that makes the function equal to zero.
f(x) = 5^(x-5) - 5 = 0
Solving for x, we get:
5^(x-5) = 5
x - 5 = 1
x = 6
Therefore, the function f(x) = 5^(x-5) - 5 has an x-intercept at x = 6.
C. f(x) = -5^(x-5) + 5
This function has a base of 5, but the exponent is negative, and the constant term is 5. To determine if it has an x-intercept, we need to find the value of x that makes the function equal to zero.
f(x) = -5^(x-5) + 5 = 0
Solving for x, we get:
-5^(x-5) = -5
5^(x-5) = 5
x - 5 = 1
x = 6
Therefore, the function f(x) = -5^(x-5) + 5 has an x-intercept at x = 6.
D. f(x) = -5^(x-5) - 1
This function has a base of 5, but the exponent is negative, and the constant term is -1. To determine if it has an x-intercept, we need to find the value of x that makes the function equal to zero.
f(x) = -5^(x-5) - 1 = 0
Solving for x, we get:
-5^(x-5) = 1
5^(x-5) = -1
There is no real value of x that satisfies this equation, as the base 5 is always positive. Therefore, the function f(x) = -5^(x-5) - 1 does not have an x-intercept.
Conclusion
In conclusion, we have analyzed four different exponential functions and determined which one does not have an x-intercept. The function f(x) = -5^(x-5) - 1 does not have an x-intercept, as there is no real value of x that satisfies the equation.
Key Takeaways
- Exponential functions are a fundamental concept in mathematics.
- X-intercepts are an essential concept in mathematics, as they help us understand the behavior of functions and their relationship with the x-axis.
- To determine if an exponential function has an x-intercept, we need to find the value of x that makes the function equal to zero.
- The function f(x) = -5^(x-5) - 1 does not have an x-intercept, as there is no real value of x that satisfies the equation.
Final Answer
Introduction
In our previous article, we explored the concept of exponential functions and x-intercepts. We analyzed four different exponential functions and determined which one does not have an x-intercept. In this article, we will provide a comprehensive Q&A guide to help you better understand the concept of exponential functions and x-intercepts.
Q: What is an exponential function?
A: An exponential function is a type of mathematical function that involves an exponent, which is a number or expression raised to a power. The general form of an exponential function is:
f(x) = a^x + b
where a is the base and b is the constant term.
Q: What is an x-intercept?
A: An x-intercept is a point on the graph of a function where the y-coordinate is zero. In other words, it is the point where the function crosses the x-axis.
Q: How do I determine if an exponential function has an x-intercept?
A: To determine if an exponential function has an x-intercept, you need to find the value of x that makes the function equal to zero. This can be done by solving the equation:
f(x) = 0
Q: What is the difference between a positive and negative exponent?
A: A positive exponent means that the base is raised to a power, while a negative exponent means that the base is raised to a negative power. For example:
f(x) = 5^x
f(x) = 5^(-x)
Q: Can an exponential function have more than one x-intercept?
A: Yes, an exponential function can have more than one x-intercept. This occurs when the function has multiple solutions to the equation f(x) = 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 create a graph.
Q: What are some common applications of exponential functions?
A: Exponential functions have many real-world applications, including:
- Population growth and decay
- Compound interest and finance
- Radioactive decay and nuclear physics
- Electrical circuits and electronics
Q: Can I use exponential functions to model real-world phenomena?
A: Yes, exponential functions can be used to model many real-world phenomena, including population growth and decay, compound interest and finance, and radioactive decay and nuclear physics.
Q: How do I choose the correct exponential function to model a real-world phenomenon?
A: To choose the correct exponential function to model a real-world phenomenon, you need to consider the following factors:
- The rate of growth or decay
- The initial value
- The time period
Q: What are some common mistakes to avoid when working with exponential functions?
A: Some common mistakes to avoid when working with exponential functions include:
- Confusing the base and exponent
- Failing to consider the domain and range of the function
- Not using the correct notation and terminology
Conclusion
In conclusion, exponential functions and x-intercepts are fundamental concepts in mathematics. By understanding these concepts, you can better analyze and solve problems involving exponential functions. We hope this Q&A guide has been helpful in providing you with a comprehensive understanding of exponential functions and x-intercepts.
Key Takeaways
- Exponential functions are a fundamental concept in mathematics.
- X-intercepts are an essential concept in mathematics, as they help us understand the behavior of functions and their relationship with the x-axis.
- To determine if an exponential function has an x-intercept, you need to find the value of x that makes the function equal to zero.
- Exponential functions have many real-world applications, including population growth and decay, compound interest and finance, and radioactive decay and nuclear physics.
Final Answer
The correct answer is D. f(x) = -5^(x-5) - 1.