Evaluate rational expressions Set 2

Learning Objectives Evaluate rational expressions for given values of multiple variables. Substitute binomial expressions into rational expressions and simplify the result. Identify values for which a rational expression is undefined, even after simplification. Simplify complex rational expressions before substituting numerical values to increase efficiency. Apply function notation to evaluate rational functions for numerical and algebraic inputs (e.g., f(3), f(x-2)). Solve real-world problems by evaluating rational expressions that model specific scenarios. Ever wondered how engineers calculate the stress on a bridge beam or how economists model market trends? 🌉 It often involves plugging complex values into rational expressions! This tutorial moves beyond simple substitu...

TermDefinitionExample Rational ExpressionA fraction where the numerator and the denominator are both polynomials. The denominator cannot be the zero polynomial.`(3x^2 - 2x + 5) / (x - 4)` Domain of a Rational ExpressionThe set of all input values (e.g., real numbers) for which the expression is defined. It excludes any values that make the denominator zero.For `(x+1)/(x-2)`, the domain is all real numbers except x=2, written as `x ∈ ℝ, x ≠ 2`. Undefined ValueA value of a variable that causes the denominator of a rational expression to equal zero, resulting in division by zero.For the expression `(y^2)/(y+3)`, the expression is undefined at `y = -3`. Substitution PrincipleThe process of replacing a variable with a specific numerical value or another algebraic expression.To evaluate `(a+b)/...

Evaluation by Direct Substitution Given `R(x) = P(x) / Q(x)` and a value `x = a`, the result is `R(a) = P(a) / Q(a)`, provided `Q(a) ≠ 0`. This is the fundamental method. Replace every instance of the variable with the given value and perform the arithmetic operations, following the order of operations (PEMDAS/BODMAS). Condition for Undefined Expressions A rational expression `R(x) = P(x) / Q(x)` is undefined for any value `x = a` such that `Q(a) = 0`. Before evaluating, always identify the values that make the denominator zero. These values are not in the domain of the expression. Evaluation by Substitution of an Expression To evaluate `R(x) = P(x) / Q(x)` for `x = g(a)`, substitute `g(a)` for every `x`: `R(g(a)) = P(g(a)) / Q(g(a))`. This is used when substituting...