Chapter 6 Rational Expressions, Functions, and Equations

§ 6.1 Rational Expressions and Functions: Multiplying and Dividing

Rational Expressions A rational expression consists of a polynomial divided by a nonzero polynomial (denominator cannot be equal to 0). A rational function is a function defined by a formula that is a rational expression. For example, the following is a rational function: Blitzer, Algebra for College Students, 6e – Slide #3 Section 6.1

Rational Expressions The rational function EXAMPLE The rational function models the cost, f (x) in millions of dollars, to inoculate x% of the population against a particular strain of flu. The graph of the rational function is shown. Use the function’s equation to solve the following problem. Find and interpret f (60). Identify your solution as a point on the graph. Blitzer, Algebra for College Students, 6e – Slide #4 Section 6.1

Rational Expressions CONTINUED Blitzer, Algebra for College Students, 6e – Slide #5 Section 6.1

Rational Expressions CONTINUED SOLUTION We use substitution to evaluate a rational function, just as we did to evaluate other functions in Chapter 2. This is the given rational function. Replace each occurrence of x with 60. Perform the indicated operations. Blitzer, Algebra for College Students, 6e – Slide #6 Section 6.1

Rational Expressions CONTINUED Thus, f (60) = 195. This means that the cost to inoculate 60% of the population against a particular strain of the flu is $195 million. The figure below illustrates the solution by the point (60,195) on the graph of the rational function. (60,195) Blitzer, Algebra for College Students, 6e – Slide #7 Section 6.1

Rational Expressions - Domain EXAMPLE Find the domain of f if SOLUTION The domain of f is the set of all real numbers except those for which the denominator is zero. We can identify such numbers by setting the denominator equal to zero and solving for x. Set the denominator equal to 0. Factor. Blitzer, Algebra for College Students, 6e – Slide #8 Section 6.1

Rational Expressions - Domain CONTINUED or Set each factor equal to 0. Solve the resulting equations. Because 4 and 9 make the denominator zero, these are the values to exclude. Thus, or Blitzer, Algebra for College Students, 6e – Slide #9 Section 6.1

Rational Expressions - Domain CONTINUED In this example, we excluded 4 and 9 from the domain. Unlike the graph of a polynomial which is continuous, this graph has two breaks in it – one at each of the excluded values. Since x cannot be 4 or 9, there is not a function value corresponding to either of those x values. At 4 and at 9, there will be dashed vertical lines called vertical asymptotes. The graph of the function will approach these vertical lines on each side as the x values draw closer and closer to each of them, but will not touch (cross) the vertical lines. The lines x = 4 and x = 9 each represent vertical asymptotes for this particular function. Blitzer, Algebra for College Students, 6e – Slide #10 Section 6.1

Horizontal Asymptotes Simplifying Rational Expressions Vertical Asymptotes A vertical line that the graph of a function approaches, but does not touch. Horizontal Asymptotes A horizontal line that the graph of a function approaches as x gets very large or very small. The graph of a function may touch/cross its horizontal asymptote. Simplifying Rational Expressions 1) Factor the numerator and the denominator completely. 2) Divide both the numerator and the denominator by any common factors. Blitzer, Algebra for College Students, 6e – Slide #11 Section 6.1

Simplifying Rational Expressions EXAMPLE Simplify: SOLUTION Factor the numerator and denominator. Divide out the common factor, x + 1. Simplify. Blitzer, Algebra for College Students, 6e – Slide #12 Section 6.1

Simplifying Rational Expressions EXAMPLE Simplify: SOLUTION Factor the numerator and denominator. Rewrite 3 – x as (-1)(-3 + x). Rewrite -3 + x as x – 3. Blitzer, Algebra for College Students, 6e – Slide #13 Section 6.1

Simplifying Rational Expressions CONTINUED Divide out the common factor, x – 3. Simplify. Blitzer, Algebra for College Students, 6e – Slide #14 Section 6.1

Multiplying Rational Expressions 1) Factor all numerators and denominators completely. 2) Divide numerators and denominators by common factors. 3) Multiply the remaining factors in the numerators and multiply the remaining factors in the denominators. Blitzer, Algebra for College Students, 6e – Slide #15 Section 6.1

Multiplying Rational Expressions EXAMPLE Multiply: SOLUTION This is the original expression. Factor the numerators and denominators completely. Divide numerators and denominators by common factors. Blitzer, Algebra for College Students, 6e – Slide #16 Section 6.1

Multiplying Rational Expressions CONTINUED Multiply the remaining factors in the numerators and in the denominators. Note that when simplifying rational expressions or multiplying rational expressions, we just used factoring. With one additional step that is provided in the following Definition for Division, division of rational expressions promises to be just as straightforward. Blitzer, Algebra for College Students, 6e – Slide #17 Section 6.1

Dividing Rational Expressions Simplifying Rational Expressions with Opposite Factors in the Numerator and Denominator The quotient of two polynomials that have opposite signs and are additive inverses is -1. Dividing Rational Expressions If P, Q, R, and S are polynomials, where then Replace with its reciprocal by interchanging its numerator and denominator. Change division to multiplication. Blitzer, Algebra for College Students, 6e – Slide #18 Section 6.1

Multiplying Rational Expressions EXAMPLE Multiply: SOLUTION This is the original expression. Factor the numerators and denominators completely. (-1) Divide numerators and denominators by common factors. Because 3 – y and y -3 are opposites, their quotient is -1. Blitzer, Algebra for College Students, 6e – Slide #19 Section 6.1

Multiplying Rational Expressions CONTINUED Now you may multiply the remaining factors in the numerators and in the denominators. or Blitzer, Algebra for College Students, 6e – Slide #20 Section 6.1

Dividing Rational Expressions EXAMPLE Divide: SOLUTION This is the original expression. Invert the divisor and multiply. Factor. Blitzer, Algebra for College Students, 6e – Slide #21 Section 6.1

Dividing Rational Expressions CONTINUED Divide numerators and denominators by common factors. Multiply the remaining factors in the numerators and in the denominators. Blitzer, Algebra for College Students, 6e – Slide #22 Section 6.1