Problem 91 - Algebra A Combined Function

Question

Fill in the table with the opposite (additive inverse), the reciprocal (multiplicative inverse), or the expression. Assume that the value of each expression is not 0. $$ \text { Reciprocal: } $$ $$ \frac{1}{2 x} $$

Step-by-Step Solution

Answer

The reciprocal of \( \frac{1}{2x} \) is \( 2x \).

Step 1: Understand What a Reciprocal Is

A reciprocal, or multiplicative inverse, of a number is another number which, when multiplied with the original, results in 1. For instance, the reciprocal of a number \( a \) is \( \frac{1}{a} \), and vice versa.

Step 2: Identify the Given Expression

The expression is \( \frac{1}{2x} \). We need to find this expression's reciprocal while assuming \( 2x eq 0 \).

Step 3: Find the Reciprocal of the Given Expression

To find the reciprocal of \( \frac{1}{2x} \), rearrange the expression to find a number that, when multiplied with \( \frac{1}{2x} \), results in 1. The reciprocal of \( \frac{1}{a} \) is \( a \), so the reciprocal of \( \frac{1}{2x} \) is \( 2x \). This is because \( \frac{1}{2x} \times 2x = 1 \).

Key Concepts

Multiplicative Inverse

In mathematics, the multiplicative inverse of a number is what you multiply that number by to get the product of 1. Often, this term goes by other names such as the reciprocal. It's important to know that this concept is applicable to fractions, integers, and algebraic expressions.
For example, if you have a simple fraction like \( \frac{3}{4} \), its reciprocal would be \( \frac{4}{3} \). This works because:

• \( \frac{3}{4} \times \frac{4}{3} = \frac{12}{12} = 1 \)

When dealing with algebraic expressions, such as \( \frac{1}{2x} \), finding the reciprocal involves flipping the fraction. Here, simply switch the numerator with the denominator. Hence, the reciprocal becomes \( 2x \). Multiplicative inverses are quite handy in solving algebraic equations, especially when isolating variables.

Additive Inverse

The additive inverse is a core concept found in algebra. It represents the number that, when added to the original, yields zero. This can easily be remembered as simply the opposite of a given value.
For any number \( a \), its additive inverse is \( -a \). When you combine these, you get zero, meaning:

• \( a + (-a) = 0 \)

This concept also extends to algebraic expressions. For instance, if you have \( 5x \), its additive inverse would be \( -5x \). Let's look at another example: if you are given \( -3 \), the additive inverse is \( 3 \), because when you sum them, they neutralize each other to produce zero.
Additive inverses are crucial for simplifying expressions and solving algebraic equations, as they often help in isolating variables by "canceling out" certain terms.

Expressions in Algebra

Algebraic expressions consist of numbers, variables, and operations like addition, subtraction, multiplication, and division. These expressions form the building blocks of algebra, allowing complex problems to be solved in a structured manner.
The first task when working with expressions is to understand the components:

• **Terms**: Parts of an expression separated by addition or subtraction, such as in \( 3x^2 + 2x + 1 \).
• **Coefficients**: Numbers that multiply the variables, like 3 and 2 in the expression above.
• **Variables**: Symbols that represent numbers, commonly \( x \) or \( y \).

Expressions can be simplified by combining like terms or factoring, making equations easier to solve. For example, in \( 2x + 3x \), you can combine the terms to say \( 5x \). Often, understanding expressions fully requires seeing the relationship between terms.
Such expressions often appear in areas like solving quadratic equations, working out word problems, and in calculus. Recognizing and manipulating these expressions is key to success in algebra and higher mathematics.