The Function $h$ Is Defined By $h(x)=3x^2-7$. Find $ H ( 5 Z ) H(5z) H ( 5 Z ) [/tex].
Introduction
In mathematics, functions play a crucial role in representing relationships between variables. The function h(x) is defined as h(x) = 3x^2 - 7. In this article, we will explore the function h(x) and find the value of h(5z).
Understanding the Function h(x)
The function h(x) is a quadratic function, which means it is a polynomial of degree two. The general form of a quadratic function is ax^2 + bx + c, where a, b, and c are constants. In this case, the function h(x) is defined as h(x) = 3x^2 - 7.
Properties of the Function h(x)
The function h(x) has several properties that make it useful in mathematics. Some of these properties include:
- Domain: The domain of a function is the set of all possible input values. In this case, the domain of h(x) is all real numbers, denoted as R.
- Range: The range of a function is the set of all possible output values. In this case, the range of h(x) is all real numbers, denoted as R.
- Even function: A function is even if f(-x) = f(x) for all x in the domain. In this case, h(-x) = 3(-x)^2 - 7 = 3x^2 - 7 = h(x), so h(x) is an even function.
Finding h(5z)
To find h(5z), we need to substitute 5z into the function h(x) in place of x. This means we will replace every instance of x with 5z.
h(5z) = 3(5z)^2 - 7
Simplifying the Expression
To simplify the expression, we need to expand the squared term.
h(5z) = 3(25z^2) - 7
Using the Distributive Property
The distributive property states that a(b + c) = ab + ac. We can use this property to simplify the expression further.
h(5z) = 75z^2 - 7
Conclusion
In this article, we defined the function h(x) as h(x) = 3x^2 - 7 and found the value of h(5z). We used the properties of the function, such as its domain and range, to understand its behavior. We also simplified the expression for h(5z) using algebraic techniques.
Real-World Applications
The function h(x) has several real-world applications, including:
- Physics: The function h(x) can be used to model the motion of an object under the influence of a quadratic force.
- Engineering: The function h(x) can be used to design and optimize systems that involve quadratic relationships.
- Economics: The function h(x) can be used to model the behavior of economic systems that involve quadratic relationships.
Final Thoughts
In conclusion, the function h(x) is a quadratic function that has several properties and applications. We found the value of h(5z) by substituting 5z into the function and simplifying the expression using algebraic techniques. The function h(x) has several real-world applications, including physics, engineering, and economics.
References
- [1] "Functions" by Khan Academy
- [2] "Quadratic Functions" by Math Open Reference
- [3] "Algebraic Techniques" by Wolfram MathWorld
The Function h(x) and Its Application in Mathematics: Q&A ===========================================================
Introduction
In our previous article, we explored the function h(x) and found the value of h(5z). In this article, we will answer some frequently asked questions about the function h(x) and its application in mathematics.
Q&A
Q: What is the domain of the function h(x)?
A: The domain of the function h(x) is all real numbers, denoted as R.
Q: What is the range of the function h(x)?
A: The range of the function h(x) is all real numbers, denoted as R.
Q: Is the function h(x) an even function?
A: Yes, the function h(x) is an even function because h(-x) = h(x) for all x in the domain.
Q: How do I find the value of h(5z)?
A: To find the value of h(5z), you need to substitute 5z into the function h(x) in place of x. This means you will replace every instance of x with 5z.
Q: Can I use the function h(x) to model real-world phenomena?
A: Yes, the function h(x) can be used to model real-world phenomena that involve quadratic relationships. Some examples include physics, engineering, and economics.
Q: How do I simplify the expression for h(5z)?
A: To simplify the expression for h(5z), you can use algebraic techniques such as expanding the squared term and using the distributive property.
Q: What are some real-world applications of the function h(x)?
A: Some real-world applications of the function h(x) include:
- Physics: The function h(x) can be used to model the motion of an object under the influence of a quadratic force.
- Engineering: The function h(x) can be used to design and optimize systems that involve quadratic relationships.
- Economics: The function h(x) can be used to model the behavior of economic systems that involve quadratic relationships.
Q: Can I use the function h(x) to solve optimization problems?
A: Yes, the function h(x) can be used to solve optimization problems that involve quadratic relationships. For example, you can use the function h(x) to find the maximum or minimum value of a quadratic function.
Q: How do I graph the function h(x)?
A: To graph the function h(x), you can use a graphing calculator or a computer algebra system. You can also use algebraic techniques to find the x-intercepts and y-intercepts of the function.
Q: Can I use the function h(x) to model periodic phenomena?
A: Yes, the function h(x) can be used to model periodic phenomena that involve quadratic relationships. For example, you can use the function h(x) to model the motion of a pendulum or a spring.
Conclusion
In this article, we answered some frequently asked questions about the function h(x) and its application in mathematics. We covered topics such as the domain and range of the function, its even property, and its real-world applications. We also provided examples of how to simplify the expression for h(5z) and how to graph the function h(x).
Final Thoughts
In conclusion, the function h(x) is a quadratic function that has several properties and applications. We hope that this article has provided you with a better understanding of the function h(x) and its role in mathematics.
References
- [1] "Functions" by Khan Academy
- [2] "Quadratic Functions" by Math Open Reference
- [3] "Algebraic Techniques" by Wolfram MathWorld