Rational functions #4 - Questions and Answers

Question 1 For the following exercises, find the domain of the rational functions. f(x) = (x² + 4) / (x² - 2x - 8) Answer Function: f(x) = (x² + 4) / (x² - 2x - 8) For a function that is defined by a fraction, the function is undefined when its denominator is equal to 0, therefore equate the denominator with 0 to evaluate the value of x for which the function will be undefined: x² - 2x - 8 = 0 Factor using ac method: x² + 2x - 4x - 8 = 0 Factor: (x+2)(x-4) = 0 Now equate each factor with 0 and solve for the variable, therefore: x = -2 and x = 4 Therefore, it can be concluded that the domain of the function are all real numbers except x = -2 and x = 4. Question 2 True or False. The domain of every rational function is the set of all real numbers. Answer 1 for example: f(x) = 1/x, the domain is not the set of all real numbers RESULT: False Answer 2 False. The domain of every rational function is the set of all real numbers, except for the values of x that make the denominator zero. Explanation: A rational function is defined as a function of the form: f(x) = p(x) / q(x) where p(x) and q(x) are polynomials and q(x) ≠ 0. The domain of a rational function is the set of all real numbers except the values of x for which the denominator q(x) is equal to zero. This is because dividing by zero is undefined in mathematics. Hence, the domain of a rational function is: Domain = {x ∈ R | q(x) ≠ 0} Therefore, the statement "The domain of every rational function is the set of all real numbers" is false. The domain consists of all real numbers except those that make the denominator zero. Answer 3 True or False. The domain of every rational function is the set of all real numbers. The correct answer is: False. The domain of a rational function is the set of all real numbers except for the values that make the denominator equal to zero. Answer 4 To determine whether the statement "The domain of every rational function is the set of all real numbers" is true or false, let's first define a rational function. A rational function is defined as the quotient of two polynomials, where the denominator polynomial is not equal to zero. Let's suppose we have a rational function f(x). The domain of f(x) consists of all the values of x for which the function is defined. In other words, it is the set of all real numbers that we can substitute into the rational function to obtain a valid output. For a rational function, the only values that we need to exclude from the domain are those that make the denominator equal to zero. This is because division by zero is undefined in mathematics. Thus, the domain of a rational function is the set of all real numbers except for the values of x that make the denominator equal to zero. We can express this using LaTeX markup as: Domain of a rational function = {x ∈ R : denominator ≠ 0} Therefore, the statement "The domain of every rational function is the set of all real numbers" is false. The domain of a rational function excludes the values of x that make the denominator equal to zero. Question 3 True or False. The graph of a rational operate could encounter a horizontal straight line. Answer 1 True, the graph of a rational function can cross a horizontal Asymptote. Answer 2 The graph of a rational function could encounter a horizontal straight line' can be represented as: True or False. The graph of a rational function could encounter a horizontal straight line? To answer this statement, we can say: False. The graph of a rational function cannot encounter a horizontal straight line. Answer 3 In mathematics, a rational function is defined as the ratio of two polynomial functions. The general form of a rational function is: f(x) = P(x) / Q(x) where P(x) and Q(x) are polynomials. Now, to determine whether the graph of a rational function can intersect a horizontal straight line, we need to consider the behavior of the function as x approaches positive or negative infinity. If the degree of the numerator polynomial P(x) is greater than or equal to the degree of the denominator polynomial Q(x), then the graph of the rational function may have a horizontal asymptote. In this case, the function approaches a constant value as x goes to infinity or negative infinity. If the degree of P(x) is less than the degree of Q(x), then the rational function may have a slant asymptote. The function approaches a linear function as x approaches infinity or negative infinity. In either case, since the function approaches a certain value or a linear function as x becomes extremely large or small, the graph of a rational function cannot intersect a horizontal straight line. The function either approaches the horizontal line or diverges away from it. Therefore, the statement 'The graph of a rational function could encounter a horizontal straight line' is false. Answer 4 True or False. The graph of a rational function could intersect a horizontal straight line. To determine whether this statement is true or false, we need to consider the characteristics of rational functions. A rational function is defined as the ratio of two polynomial functions, where the denominator is not equal to zero. The general form of a rational function is given by: f(x) = P(x) / Q(x) where P(x) and Q(x) are polynomials. In the graph of a rational function, the vertical asymptotes occur where the denominator Q(x) is equal to zero. However, there are no restrictions on the horizontal behavior of the graph. Therefore, it is true that the graph of a rational function could intersect a horizontal straight line. True Question 4 Is -18/6 irrational, rational, natural, or integer Answer Given -18/6 Now, -18/6 = -3 which is rational number and also an integer Therefore, -18/6 is rational and also integer Question 5 Explain how the sets of numbers (counting, whole, integer, rational, irrationals, reals) are related to each other. Answer Concept: A branch of mathematics which deals with symbols and the rules for manipulating those symbols is called algebra. Those symbols which represent quantities without fixed values known as variables. Given: The objective is to give an explanation to the set of counting, whole, integer, rational, irrational, real numbers and the related to each other Thus all the counting numbers are whole numbers where, W = 0,1,2,3,... the counting numbers are 1,2,3,... The integers are whole numbers ...-3,-2,-1,0,1,2,3,... The rational numbers are quotients of the integer 1/2,6,6/7,.... Irrational number are that where the it is not possible to right the quotients of the integer π = √2 It known that the real numbers are either rational or irrational numbers and the real numbers are the larger set of rational or irrational numbers where rational and the irrational numbers are subsets. A subset of rational numbers are subsets whereas for integer it is whole number and the subset for whole number is counting numbers.