Differentiate The Function.$\[ Y = \frac{1}{(7x - 2)^2} \\]
Introduction
In mathematics, functions play a crucial role in describing the relationship between variables. The given function, y = 1 / (7x - 2)^2, is a rational function that involves a quadratic expression in the denominator. In this article, we will delve into the properties and behavior of this function, exploring its domain, range, and key characteristics.
Domain and Range
The domain of a function is the set of all possible input values for which the function is defined. In the case of the given function, the denominator (7x - 2)^2 cannot be equal to zero, as division by zero is undefined. Therefore, we must find the values of x that make the denominator zero.
To find the values of x that make the denominator zero, we set the expression (7x - 2)^2 equal to zero and solve for x.
(7x - 2)^2 = 0
Taking the square root of both sides, we get:
7x - 2 = 0
Adding 2 to both sides, we get:
7x = 2
Dividing both sides by 7, we get:
x = 2/7
Therefore, the value of x that makes the denominator zero is x = 2/7. This value must be excluded from the domain of the function.
The domain of the function is all real numbers except x = 2/7.
The range of a function is the set of all possible output values. Since the function is a rational function with a quadratic expression in the denominator, its range is all positive real numbers.
Key Characteristics
The given function has several key characteristics that are worth exploring.
- Asymptotes: The function has a vertical asymptote at x = 2/7, which is the value of x that makes the denominator zero. As x approaches 2/7 from the left, the function approaches negative infinity. As x approaches 2/7 from the right, the function approaches positive infinity.
- Holes: The function has a hole at x = 2/7, which is the value of x that makes the denominator zero. This hole is a removable discontinuity, meaning that the function can be made continuous at this point by redefining the function at this point.
- End behavior: As x approaches positive or negative infinity, the function approaches zero.
Graphing the Function
The graph of the function is a curve that approaches the vertical asymptote at x = 2/7. The graph has a hole at x = 2/7, which is a removable discontinuity. As x approaches positive or negative infinity, the graph approaches the x-axis.
Conclusion
In conclusion, the function y = 1 / (7x - 2)^2 is a rational function with a quadratic expression in the denominator. Its domain is all real numbers except x = 2/7, and its range is all positive real numbers. The function has a vertical asymptote at x = 2/7, a hole at x = 2/7, and end behavior that approaches zero as x approaches positive or negative infinity.
Applications
The function y = 1 / (7x - 2)^2 has several applications in mathematics and science.
- Calculus: The function is used in calculus to model the behavior of functions with asymptotes and holes.
- Physics: The function is used in physics to model the behavior of particles with asymptotes and holes.
- Engineering: The function is used in engineering to model the behavior of systems with asymptotes and holes.
Real-World Examples
The function y = 1 / (7x - 2)^2 has several real-world examples.
- Optics: The function is used in optics to model the behavior of light with asymptotes and holes.
- Electronics: The function is used in electronics to model the behavior of circuits with asymptotes and holes.
- Biology: The function is used in biology to model the behavior of populations with asymptotes and holes.
Conclusion
Q: What is the domain of the function y = 1 / (7x - 2)^2?
A: The domain of the function is all real numbers except x = 2/7. This is because the denominator (7x - 2)^2 cannot be equal to zero, as division by zero is undefined.
Q: What is the range of the function y = 1 / (7x - 2)^2?
A: The range of the function is all positive real numbers. This is because the function is a rational function with a quadratic expression in the denominator, and the numerator is always positive.
Q: What is the vertical asymptote of the function y = 1 / (7x - 2)^2?
A: The vertical asymptote of the function is x = 2/7. This is because the denominator (7x - 2)^2 approaches zero as x approaches 2/7 from the left, and the function approaches negative infinity. As x approaches 2/7 from the right, the function approaches positive infinity.
Q: What is the hole in the function y = 1 / (7x - 2)^2?
A: The hole in the function is at x = 2/7. This is because the function has a removable discontinuity at this point, meaning that the function can be made continuous at this point by redefining the function at this point.
Q: What is the end behavior of the function y = 1 / (7x - 2)^2?
A: As x approaches positive or negative infinity, the function approaches zero. This is because the denominator (7x - 2)^2 approaches infinity as x approaches positive or negative infinity, and the function approaches zero.
Q: How is the function y = 1 / (7x - 2)^2 used in calculus?
A: The function is used in calculus to model the behavior of functions with asymptotes and holes. It is also used to study the properties of functions, such as their domain, range, and continuity.
Q: How is the function y = 1 / (7x - 2)^2 used in physics?
A: The function is used in physics to model the behavior of particles with asymptotes and holes. It is also used to study the properties of particles, such as their energy and momentum.
Q: How is the function y = 1 / (7x - 2)^2 used in engineering?
A: The function is used in engineering to model the behavior of systems with asymptotes and holes. It is also used to study the properties of systems, such as their stability and response to inputs.
Q: What are some real-world examples of the function y = 1 / (7x - 2)^2?
A: Some real-world examples of the function include:
- Optics: The function is used in optics to model the behavior of light with asymptotes and holes.
- Electronics: The function is used in electronics to model the behavior of circuits with asymptotes and holes.
- Biology: The function is used in biology to model the behavior of populations with asymptotes and holes.
Q: How can I graph the function y = 1 / (7x - 2)^2?
A: You can graph the function using a graphing calculator or a computer algebra system. You can also use a graphing software or a programming language to create a graph of the function.
Q: How can I find the values of x that make the denominator (7x - 2)^2 equal to zero?
A: To find the values of x that make the denominator (7x - 2)^2 equal to zero, you can set the expression (7x - 2)^2 equal to zero and solve for x. This will give you the values of x that make the denominator zero, and you can exclude these values from the domain of the function.