Properties of the Reciprocal Function

Oct 13, 2025 by miodeya.com 68 views

The reciprocal function, denoted as $f(x) = \frac{1}{x}$ , is a fundamental concept in mathematics that plays a crucial role in various branches of mathematics, including algebra, calculus, and analysis. In this article, we will delve into the properties of the reciprocal function and explore which of the given points lies on its graph.

Properties of the Reciprocal Function

The reciprocal function $f(x) = \frac{1}{x}$ has several key properties that are essential to understand its behavior. Firstly, the function is defined for all real numbers except $x = 0$ , as division by zero is undefined. This implies that the domain of the reciprocal function is the set of all real numbers except zero.

Secondly, the reciprocal function is a continuous function, meaning that it can be drawn without lifting the pencil from the paper. This continuity is a result of the fact that the function is defined for all real numbers except zero, and the limit of the function as $x$ approaches zero is infinity.

Lastly, the reciprocal function is an odd function, meaning that $f(-x) = -f(x)$ . This property can be verified by substituting $-x$ into the function and simplifying.

Graph of the Reciprocal Function

The graph of the reciprocal function $f(x) = \frac{1}{x}$ is a hyperbola that approaches the x-axis as $x$ approaches infinity and the y-axis as $x$ approaches negative infinity. The graph has a vertical asymptote at $x = 0$ , as the function is undefined at this point.

To determine which of the given points lies on the graph of the reciprocal function, we need to substitute each point into the function and verify if the resulting value is equal to the y-coordinate of the point.

Analyzing the Given Points

Let's analyze each of the given points and determine if it lies on the graph of the reciprocal function.

A. ${$ (-21, -\frac{1}{21})$}$

To verify if this point lies on the graph of the reciprocal function, we substitute $x = -21$ into the function and simplify:

$f(-21) = \frac{1}{-21} = -\frac{1}{21}$

Since the resulting value is equal to the y-coordinate of the point, this point lies on the graph of the reciprocal function.

B. ${$ (\frac{3}{4}, \frac{3}{4})$}$

To verify if this point lies on the graph of the reciprocal function, we substitute $x = \frac{3}{4}$ into the function and simplify:

$f(\frac{3}{4}) = \frac{1}{\frac{3}{4}} = \frac{4}{3}$

Since the resulting value is not equal to the y-coordinate of the point, this point does not lie on the graph of the reciprocal function.

C. ${$ (\frac{1}{21}, -21)$}$

To verify if this point lies on the graph of the reciprocal function, we substitute $x = \frac{1}{21}$ into the function and simplify:

$f(\frac{1}{21}) = \frac{1}{\frac{1}{21}} = 21$

Since the resulting value is not equal to the y-coordinate of the point, this point does not lie on the graph of the reciprocal function.

D. ${$ (\frac{5}{7}, \frac{7}{5})$}$

To verify if this point lies on the graph of the reciprocal function, we substitute $x = \frac{5}{7}$ into the function and simplify:

$f(\frac{5}{7}) = \frac{1}{\frac{5}{7}} = \frac{7}{5}$

Since the resulting value is not equal to the y-coordinate of the point, this point does not lie on the graph of the reciprocal function.

Conclusion

In conclusion, the point ${$ (-21, -\frac{1}{21})$}$ lies on the graph of the reciprocal function $f(x) = \frac{1}{x}$ . This is because the resulting value of substituting $x = -21$ into the function is equal to the y-coordinate of the point.

The other points, ${$ (\frac{3}{4}, \frac{3}{4})$}$, ${$ (\frac{1}{21}, -21)$}$, and ${$ (\frac{5}{7}, \frac{7}{5})$}$, do not lie on the graph of the reciprocal function, as the resulting values of substituting their x-coordinates into the function are not equal to their y-coordinates.

References

In this article, we will answer some frequently asked questions about the reciprocal function, including its properties, graph, and applications.

Q: What is the reciprocal function?

A: The reciprocal function, denoted as $f(x) = \frac{1}{x}$ , is a fundamental concept in mathematics that plays a crucial role in various branches of mathematics, including algebra, calculus, and analysis.

Q: What are the properties of the reciprocal function?

A: The reciprocal function has several key properties, including:

The function is defined for all real numbers except $x = 0$ , as division by zero is undefined.
The function is a continuous function, meaning that it can be drawn without lifting the pencil from the paper.
The function is an odd function, meaning that $f(-x) = -f(x)$ .

Q: What is the graph of the reciprocal function?

A: The graph of the reciprocal function $f(x) = \frac{1}{x}$ is a hyperbola that approaches the x-axis as $x$ approaches infinity and the y-axis as $x$ approaches negative infinity. The graph has a vertical asymptote at $x = 0$ , as the function is undefined at this point.

Q: How do I determine if a point lies on the graph of the reciprocal function?

A: To determine if a point lies on the graph of the reciprocal function, you need to substitute the x-coordinate of the point into the function and verify if the resulting value is equal to the y-coordinate of the point.

Q: What are some common applications of the reciprocal function?

A: The reciprocal function has several common applications in various fields, including:

Algebra: The reciprocal function is used to solve equations and inequalities involving fractions.
Calculus: The reciprocal function is used to find the derivative and integral of functions involving fractions.
Analysis: The reciprocal function is used to study the properties of functions involving fractions.

Q: Can I use the reciprocal function to solve real-world problems?

A: Yes, the reciprocal function can be used to solve real-world problems involving fractions, such as:

Finance: The reciprocal function is used to calculate interest rates and investment returns.
Science: The reciprocal function is used to model the behavior of physical systems involving fractions, such as the motion of objects and the flow of fluids.

Q: How do I graph the reciprocal function?

A: To graph the reciprocal function, you can use a graphing calculator or software, such as Desmos or GeoGebra. You can also use a pencil and paper to draw the graph by hand.

Q: What are some common mistakes to avoid when working with the reciprocal function?

A: Some common mistakes to avoid when working with the reciprocal function include:

Dividing by zero, which is undefined.
Not considering the domain of the function, which is all real numbers except zero.
Not using the correct notation, such as $f(x) = \frac{1}{x}$ instead of $f(x) = x^{-1}$ .

Q: Can I use the reciprocal function to solve equations involving fractions?

A: Yes, the reciprocal function can be used to solve equations involving fractions. For example, to solve the equation $\frac{1}{x} = \frac{1}{2}$ , you can multiply both sides by $x$ to get $1 = \frac{x}{2}$ , and then multiply both sides by $2$ to get $2 = x$ .

Q: How do I find the derivative of the reciprocal function?

A: To find the derivative of the reciprocal function, you can use the power rule of differentiation, which states that if $f(x) = x^n$ , then $f'(x) = nx^{n-1}$ . In this case, $f(x) = \frac{1}{x} = x^{-1}$ , so $f'(x) = -x^{-2} = -\frac{1}{x^2}$ .

Q: How do I find the integral of the reciprocal function?

A: To find the integral of the reciprocal function, you can use the power rule of integration, which states that if $f(x) = x^n$ , then $\int f(x) dx = \frac{x^{n+1}}{n+1} + C$ . In this case, $f(x) = \frac{1}{x} = x^{-1}$ , so $\int f(x) dx = \int x^{-1} dx = \ln|x| + C$ .

Conclusion

In conclusion, the reciprocal function is a fundamental concept in mathematics that plays a crucial role in various branches of mathematics, including algebra, calculus, and analysis. It has several key properties, including being defined for all real numbers except zero, being a continuous function, and being an odd function. The graph of the reciprocal function is a hyperbola that approaches the x-axis as $x$ approaches infinity and the y-axis as $x$ approaches negative infinity. The reciprocal function has several common applications in various fields, including algebra, calculus, and analysis.