ࡱ> uw rstR_] *bjbj -x-\x-\ rrqt.LzzzN \\\sssssss$)vxt\W,\\\trrzN 7+tbbb\"rzN sb\sbboh"VpN N]osAt0qtoy^y VpVp2ysT\\b\\\\\tt`\\\qt\\\\y\\\\\\\\\> 8: 7.1 Rational Expressions and Functions; Multiplying and Dividing A rational expression or algebraic fraction, is the _____________of two polynomials, again with the denominator ______ _____. For example: are all rational expressions. Rational expressions are elements of the set  SHAPE \* MERGEFORMAT  A function that is defined by a quotient of polynomials is called a rational function and has the form The ___________ of the rational function consists of all real numbers except those that make ______________ - that is, the denominatorequal to ______. For example, the domain of includes all real numbers except 5, because 5 would make the denominator equal to 0. EXAMPLE 1 For each rational function, find all numbers that are not in the domain. Then state the domain in set notation. a. b. Writing a Rational Expression in Lowest Terms Step 1 ____________ both the numerator and denominator to find their _________________. Step 2 ______________ the fundamental property. EXAMPLE 2 Write each rational expression in lowest terms. a. b. c.  SHAPE \* MERGEFORMAT  EXAMPLE 3 Write each rational expression in lowest terms. a. b. Quotient of Opposites In general, if the _____________and the _______________r of a rational expression are ____________, then the expression equals 1.  SHAPE \* MERGEFORMAT   SHAPE \* MERGEFORMAT  Multiplying Rational Expressions Step 1 ______________ all numerators and denominators as completely as possible. Step 2 Apply the fundamental property. Step 3 __________ remaining factors in the numerator and remaining factors in the denominator. Leave the ____________________ in factored form. Step 4 Check to be sure that the ______________is in ________________ terms. EXAMPLE 4 Multiply. a. b. Finding the Reciprocal To find the reciprocal of a nonzero rational expression, invert the rational expression. Reciprocals have a product of ____________. Recall that _____________ has no reciprocal. Dividing Rational Expressions To divide two rational expressions, _____________ the first (the dividend) by the _____________________________________________ (the divisor). EXAMPLE 5 Divide. a. b. 7.2 Adding and Subtracting Rational Expressions Step 1 If the denominators are the _______________ add or subtract the numerators. Place the ______________ over the common denominator. If the denominators are _______________, first find the _____________ ______________ _________________. Write all rational expressions with this least common denominator, and then add or subtract the numerators. Place the result over the common denominator. Step 2 Simplify. Write all answers in _______________________________ EXAMPLE 1 Add or subtract as indicated. a. b. c. Finding the Least Common Denominator Step 1 Factor each ________________. Step 2 Find the least common denominator. The LCD is the _________________of all of the different factors from each denominator, with each factor raised to the greatest power that occurs in any denominator. EXAMPLE 2 Add or subtract as indicated. A. B. EXAMPLE 3 Subtract. A. B. EXAMPLE 4 Add: EXAMPLE 5 Add and subtract as indicated.  EXAMPLE 6 Subtract. EXAMPLE 7 Add.  7.4 Equations with Rational Expressions and Graphs A rational equation is an equation that contains __________________________ rational expression with a _____________________ in the denominator. The easiest way to solve most rational equations is to ________________________ in the equation by the least common denominator. This step will ____________________ the equation of ______________________. It is necessary to either check the proposed solutions or verify that they are in the domain. CAUTION When both sides of an equation are multiplied by a variable expression, the resulting proposed solutions may not satisfy the original equation. You must either determine and observe the domain or check all proposed solutions in the original equation. It is wise to do both. Agreement on Domain Unless specified otherwise, the _________ of a relation is assumed to be ________________ that produce real numbers when substituted for the _____________ variable. Domain Restrictions 1. Variable under the ________________ - set whats under the _____________________ and _____________ 2. Variable is in the _______________________ - set the ____________________ and ______________ 3. In a fraction, when factored, something cancels from the numerator to the denominator leaving a ________ in the graph. (this will appear in later chapters) EXAMPLE 1 Solve. EXAMPLE 2 Solve.  Note: Since the proposed solution is not in the domain, it cannot be an actual solution of the equation. Substituting 1 into the original equation shows why. Division by 0 is undefined. The equation has no solution and the solution set is (. EXAMPLE 3 Solve. EXAMPLE 4 Solve.  12.4 Rational Functions A rational function is a function in the form , where p and q are polynomial functions and q is not the zero polynomial. Rational Functions different from other functions because they have ___________________ Asymptote -- a line that the graph of a function gets _______________ and closer to as one travels along that line in _______________- direction Asymptotes Vertical Asymptote set bottom _____________-, and solve. Same as restrictions on ______________ Form _______________ Note: The graph will ________________ cross or touch a vertical asymptote. Domain: all ___________ except when there are _______________, written in ___________________ EXAMPLE: Find the vertical asymptotes and state the domain, if any, of the graph of each rational function.  HORIZONTAL ASYMPTOTES - determined by the degrees of the numerator and denominator Form _________________ If degree of P(x) < degree of Q(x) (Top < Bottom) horizontal asymptote ____________ (x-axis). If degree of P(x) = degree of Q(x) (Top = Bottom) horizontal asymptote y = If degree of P(x) > degree of Q(x), (Top > Bottom) _____________ horizontal asymptote Range: all _______________except when there are ___________________, written in _____________ EXAMPLE: Find the horizontal asymptotes and state the range, if any, of the graph of each rational function. Remember The graph of a rational function ____________r crosses a ____________l asymptote. The graph of a rational function ______________ a _________________ asymptote, but does not necessarily do so. Steps for Graphing Set bottom = 0, solve. This gives restrictions on domain and vertical asymptotes. Set top = 0, solve. This gives x-intercepts Sub 0s in all xs. This gives y-intercepts Check exponents for horizontal asymptotes. 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