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Analysis of the Rational Algebraic Expression $\\frac{1}{x}$: Domain and Classification
Mathematics
Grade 8 (Junior High School)
Question Content
Identify or perform operations related to the algebraic expression $\\frac{1}{x}$ (the problem context is implied to be about rational expressions, as only the expression $\\frac{1}{x}$ is provided)
Correct Answer
This is a rational algebraic expression; domain: all real numbers except x=0
Detailed Solution Steps
1
Step 1: Classify the expression: $\\frac{1}{x}$ is a rational expression, which is a ratio of two polynomials where the numerator is the constant polynomial 1 and the denominator is the monomial x.
2
Step 2: Determine the valid domain: Since division by zero is undefined in mathematics, set the denominator not equal to zero, so $x \\neq 0$. The domain is all real numbers except 0.
3
Step 3: If simplification is required: The expression $\\frac{1}{x}$ is already in its simplest form as there are no common factors between the numerator and denominator to cancel out.
Knowledge Points Involved
1
Rational Expressions
A rational expression is a fraction where both the numerator and denominator are polynomials, and the denominator cannot be zero. Expressions like $\\frac{1}{x}$ fit this definition, and they follow similar arithmetic rules to numerical fractions, including addition, subtraction, multiplication, and division.
2
Domain of Algebraic Expressions
The domain of an algebraic expression is the set of all real numbers that can be substituted for the variable without making the expression undefined. For rational expressions, this means excluding any values that would make the denominator equal to zero, so for $\\frac{1}{x}$, $x$ cannot be 0.
3
Simplification of Rational Expressions
Simplifying a rational expression involves canceling out common factors between the numerator and denominator. Since the numerator 1 has no factors other than 1 and -1, $\\frac{1}{x}$ is already in its simplest simplified form with no common factors to remove.
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