Chapter 5B - Rational Functions

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Randolf Phillips
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1 Fry Texas A&M University Math 150 Chapter 5B Fall Chapter 5B - Rational Functions Definition: A rational function is The domain of a rational function is all real numbers, except those values where the denominator is 1. g(x) = 1 x +1 A fraction is zero when its is zero and its is NOT zero. That is why the graph of 1 x +1 never the Vertical asymptotes occur where the is zero, but the is not zero. e) Solve g(x) > 0 f) for large x, g(x) acts like, (This is the quotient of the.) so as x, g(x) --> and as x g(x) -->

2 Fry Texas A&M University Math 150 Chapter 5B Fall h(x) = x x +1 e) Solve h(x) > 0 f) for large x, (think about estimating) h(x) acts like, (This is the quotient of the ) so as x, h(x) --> and as x h(x) -->

3 Fry Texas A&M University Math 150 Chapter 5B Fall p(x) = x2 +1 x +1 e) Solve p(x) > 0 f) For large x, (think about estimating) p(x) acts like, (This is the quotient of the ) so as x, p(x) --> and as x p(x) -->

4 Fry Texas A&M University Math 150 Chapter 5B Fall A hole in the graph indicates a place where the function is undefined, but the function s behavior is not asymptotic. Understanding Holes in Graphs: 4. f (x) = x2 1 x +1! e) there is a hole at f) the y - coordinate of hole g) Solve f (x) > 0 h) For large x, f (x) acts like, so as x, f (x) --> and as x f (x) -->

5 Fry Texas A&M University Math 150 Chapter 5B Fall g(x) = x2 1 x 2 +1! e) there is a hole at f) the y - coordinate of hole g) Solve g(x) > 0 h) For large x, g(x) acts like, so as x, g(x) --> and as x g(x) -->

6 Fry Texas A&M University Math 150 Chapter 5B Fall x h(x) = 3( x 2 9)(x 1)! e) there is a hole at f) the y - coordinate of hole g) Solve h(x) > 0 h) for large x, h(x) acts like, so as x, h(x) --> and as x h(x) -->

7 Fry Texas A&M University Math 150 Chapter 5B Fall p(x) = 2x 3 + 6x 2 x 3 + 3x 2 4x 12 e) there is a hole at f) the y - coordinate of hole g) Solve p(x) > 0 h) For large x, p(x) acts like, so as x, p(x) --> and as x p(x) -->

8 Fry Texas A&M University Math 150 Chapter 5B Fall To graph a rational function, a) Evaluate the function at x = 0, this is the b) Find the values for x for which the numerator is zero, but the denominator is not zero. This is where the graph c) Find the values for x for which the denominator is zero, but the numerator is not zero. This is where the graph d) Find the values of x for which both the numerator and the denominator are zero. This is where there is e) To find the y - coordinate of the hole: If there is a hole at x = a, then ( x a) is a factor of both the numerator and the denominator. The rational function f (x) can be written in the form (x a) f (x) = p(x) ( x a) q(x). It could be that p(x) = 1 and/or q(x) = 1.! Let f (x) = p(x) q(x), then the y - coordinate of the hole is f a ( ). f) Simplify the quotient of the leading terms of the numerator and the denominator. The end behavior of this function is the same as the end behavior of the given function. g) Determine where the function is greater than 0.! This is where the graph of the function is.

9 Fry Texas A&M University Math 150 Chapter 5B Fall The Generalized Technique for Determining End Behavior of Rational Functions: (You WILL be asked to demonstrate your ability to use this technique on Exam 3.) 8. Use the Generalized Technique for Determining End Behavior to determine the end behavior of f (x) = x x 4 Step 1: Determine the highest power of x involved in the function. In this case it is. Step 2: Multiply the rational function by 1 in the special form: Step 3: Simplify each term. Step 4: Examine the end behavior of each term. Step 5: Use this information to determine the end behavior of the rational function. As x, f (x). So the graph of f (x) has a horizontal asymptote of

10 Fry Texas A&M University Math 150 Chapter 5B Fall Use the Generalized Technique for Determining End Behavior to determine the end behavior of g(x) = 5x x 3 Step 1: Determine the highest power of x involved in the function. In this case it is. Step 2: Multiply the rational function by 1 in the special form: Step 3: Simplify each term. Step 4: Examine the end behavior of each term. Step 5: Use this information to determine the end behavior of the rational function. As x, g(x). So the graph of g(x) has a horizontal asymptote of Extra Problems:! Text:! 1-3, 5-26!! 1 g(x) = 8x5 3 7x 4 1!! as x, g(x) --> 2. h(x) = 8x4 3 7x 5 1!! as x, h(x) -->

11 Fry Texas A&M University Math 150 Chapter 5B Fall h(x) = x 2 (1 x 3 ) As x, h(x) 4. p(x) = (1 x3 ) x 2! As x, p(x) 5. r(x) = x 2 (1 x 2 ) As x, r(x) 6. Determine the x intercept(s) of f (x) = 1 x x + 8 Intercepts are points so, if there are any, state their x and y coordinates.. If there are none, write NONE.!!!!!!!

12 Fry Texas A&M University Math 150 Chapter 5B Fall Let f (x) = x2 + x 6 x 2 3x + 2. Determine the following. If there are none, write NONE. State the intervals in interval notation. State the equation of lines. State both the x and y coordinates of the points. a) domain!!! b) y - intercept(s)!! c) x - intercept(s) d) hole(s)!! e) vertical asymptote(s) 8. f (x) = x2 8x +16 9(x 2 16) a) (3 points) List the coordinates of all of the x -intercepts.! If there are none, write NONE in the blank provided. b) (3 points) List the equations of all of the vertical asymptotes.! If there are none, write NONE in the blank provided. c) (3 points) List the equations of all of the horizontal asymptotes.! If there are none, write NONE in the blank provided. d) (4 points) List the coordinates of all the holes.! If there are none, write NONE in the blank provided.

Section Properties of Rational Expressions

Section Properties of Rational Expressions 88 Section. - Properties of Rational Expressions Recall that a rational number is any number that can be written as the ratio of two integers where the integer in the denominator cannot be. Rational Numbers: