Chapter 1 - Introduction to Algebraic Expressions - 1.7 Multiplication and Division of Real Numbers - 1.7 Exercise Set - Page 59: 128

Yes, the opposites are: $-x$ and $-\frac{1}{x}=\frac{1}{-x}$

The two numbers $x$ and $\frac{1}{x}$ are reciprocals of each other. Their opposites are: $-x$ and $-\frac{1}{x}=\frac{1}{-x}$ They are also reciprocals of each other.