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Questions and Answers
What is a rational expression?
What is a rational expression?
A rational expression is a fraction where the numerator and the denominator are polynomials and the denominator is not equal to zero.
What defines a rational equation?
What defines a rational equation?
A rational equation is an equation that contains one or more rational expressions.
What is a rational inequality?
What is a rational inequality?
A rational inequality is an inequality that contains rational expressions.
What is the form of a polynomial function?
What is the form of a polynomial function?
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What is the condition for a linear function to be constant?
What is the condition for a linear function to be constant?
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What is a rational function?
What is a rational function?
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What determines the domain of a rational function?
What determines the domain of a rational function?
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What is the restricted value in the function $f(x) = \frac{2x + 5}{x - 6}$?
What is the restricted value in the function $f(x) = \frac{2x + 5}{x - 6}$?
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What is the domain of the function $f(x) = \frac{x + 3}{x - 5}$?
What is the domain of the function $f(x) = \frac{x + 3}{x - 5}$?
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What values restrict the domain of the function $f(x) = \frac{x^2 - 9}{x + 3}$?
What values restrict the domain of the function $f(x) = \frac{x^2 - 9}{x + 3}$?
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Study Notes
Rational Equations and Inequalities
- Rational Expression: A fraction of polynomial expressions, formatted as ( \frac{A}{B} ) where ( A ) and ( B ) are polynomials and ( B \neq 0 ). Example: ( \frac{ab}{7 - x} ).
- Rational Equation: An equation that includes at least one rational expression. Example: ( \frac{ab}{7 - x} = x ).
- Rational Inequality: Involves rational expressions combined with inequality symbols (≤, ≥). Example: ( \frac{ab}{7 - x} > 2x ).
Rational Functions
- Polynomial Function: Defined by ( f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_0 ), where ( a_n \neq 0 ) and ( n ) is a non-negative integer.
- Linear Function: A special polynomial function of degree 1, expressed as ( f(x) = mx + b ), where ( m ) can be zero.
- Quadratic Function: A polynomial of degree 2, formatted as ( f(x) = ax^2 + bx + c ) with ( a \neq 0 ).
- Rational Function: Expressed as ( f(x) = \frac{N(x)}{D(x)} ) where both ( N(x) ) and ( D(x) ) are polynomials and ( D(x) ) is not the zero polynomial.
- Domain of a Rational Function: Comprises all values of ( x ) that do not result in a zero denominator.
Finding the Domain of Functions
- To determine the domain, identify values of ( x ) that cause the denominator to equal zero and exclude them.
- Examples:
- For ( f(x) = \frac{2x + 5}{x - 6} ): The restricted value is ( x = 6 ). Domain is all real numbers except ( 6 ).
- For ( f(x) = \frac{x + 3}{x + 3} ): The restricted value is ( x = -3 ). Domain is all real numbers except ( -3 ).
- For ( f(x) = \frac{x^2 - 9}{1} ): The restricted values are ( x = 3 ) and ( x = -3 ). Domain excludes ( ±3 ).
Activity
- Students are instructed to find the domain of specified functions on a whole sheet of paper and show their solutions.
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Description
Test your understanding of rational equations, inequalities, and functions. This quiz covers definitions and examples of rational expressions, equations, and various polynomial functions including linear and quadratic types. Challenge yourself to apply these concepts in problem-solving scenarios.
