Reciprocal

Definition

The reciprocal of a number is the value obtained by dividing 1 by that number. It represents the inverse relationship between two quantities, where one quantity is the multiplicative inverse of the other.

5 Must Know Facts For Your Next Test

1. The reciprocal of a number is denoted by the $\frac{1}{x}$ notation, where $x$ is the original number.
2. The reciprocal of a rational expression is obtained by interchanging the numerator and denominator.
3. Reciprocals are important in the study of rational functions, as they are used to describe the behavior and properties of these functions.
4. The graph of the reciprocal function $f(x) = \frac{1}{x}$ is a hyperbola that passes through the origin and has vertical and horizontal asymptotes.
5. Reciprocals are used to simplify rational expressions and perform operations such as multiplication and division.

Review Questions

• Explain how the reciprocal of a number is related to the concept of a multiplicative inverse.
  • The reciprocal of a number is the multiplicative inverse of that number. This means that when a number is multiplied by its reciprocal, the result is 1. For example, the reciprocal of 5 is $\frac{1}{5}$, and $5 \times \frac{1}{5} = 1$. This inverse relationship is crucial in the simplification and manipulation of rational expressions, as well as in the study of rational functions.
• Describe the graph of the reciprocal function $f(x) = \frac{1}{x}$ and explain its key features.
  • The graph of the reciprocal function $f(x) = \frac{1}{x}$ is a hyperbola that passes through the origin and has vertical and horizontal asymptotes. The vertical asymptote occurs at $x = 0$, as the function is undefined at this point. The horizontal asymptotes occur at $y = 0$ and $y = -0$, indicating that as $x$ approaches positive or negative infinity, the function approaches 0 from above and below, respectively. The shape and properties of the reciprocal function are crucial in understanding the behavior of rational functions.
• Analyze how the reciprocal of a rational expression can be used to simplify and perform operations on the expression.
  • The reciprocal of a rational expression is obtained by interchanging the numerator and denominator. This property is used to simplify rational expressions by identifying common factors in the numerator and denominator, which can then be canceled out. Additionally, the reciprocal of a rational expression is essential in performing operations such as multiplication and division, as the reciprocal can be used to convert division into multiplication. For example, to divide two rational expressions, you can multiply the first expression by the reciprocal of the second expression, effectively converting the division into a multiplication operation.

Related terms

Inverse Function:

A function that reverses the operation of another function, such that the composition of the two functions results in the identity function.

Multiplicative Inverse: A number that, when multiplied by another number, results in a product of 1.

Rational Expression:

An expression that can be written as the quotient of two polynomial expressions.