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MAC1147 Lecture Notes - Lecture 7: Real Number, Even And Odd Functions

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28 Feb 2017

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Miami Dade College

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Mathematics: Calculus & Pre-Calculus

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4. 1 polynomial functions and models notes: sterling. Provide a generalization to the terms listed in this section. Identifying polynomial functions f (x) = anxn + an 1xx 1 + + a1x + a0. Graph : horizontal line with c as y intercept. Graph : a nonhorizontal and nonvertical line with a slope and y intercept. Power function f (x) = axn a 6= 0 n > 0. Zero of multiplicity r when the following: (x r)m is a factor of f, (x r)m+1 is not a factor of f. Zero of even multiplicity f (x)"s sign doesn"t change from one side of r to the other side, which will make the graph touch the x-axis at r. Zero of odd multiplicity f (x)"s sign does change from one side of r to the other side, which will make the graph cross the x-axis at r.

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Related Questions

Algebra

Complete the sentence below. If r is a real zero of even multiplicity of a function f then the graph of f the x-axis at r. If r is a real zero of even multiplicity of a function f then the graph of f the x-axis at r. Complete the sentence below. The graphs of power functions of the form f(x) = x n: where n is an even integer, always contain the points and The graphs of power functions of the form f(x) = x n: where n is an even integer, always contain the points and If r is a solution to the equation f(x) = 05 name three additional statements that can be made about f and r assuming f is a polynomial function. Choose the correct answer below. If r is a solution to the equation f(x) = 0, then r is a real zero of the polynomial function r is an x-intercept of the graph of and x - r is a factor of f. If r is a solution to the equation f(x) = 0, then r is a complex zero of the polynomial function r is an x-intercept of the graph of and x + r is a factor of f. If r is a solution to the equation f(x) = 0, then r is a real zero of the polynomial function r is a y-intercept of the graph of x and x + r is a factor of f. Explain what the notation Km f(x) = -infinity means. x rightarrow infinity Choose the correct answer below. The limit of f(x) as x approaches - infinity equals infinity. The limit of x as f(x) approaches infinity equals - infinity. The limit of f(x) as x approaches infinity equals - infinity. The limit of x as f(x) approaches - infinity equals infinity. Form a polynomial whose zeros and degree are given. Zeros: -7, multiplicity 1; - 3, multiplicity 2; degree 3 Type a polynomial with integer coefficients and a leading coefficient of 1 in the box below. F(x) = Find the vertical horizontal and oblique asymptotes, if any, for the following rational function.R(x) = 12x / x + 20 Select the correct choice below and fill in any answer boxes within your choice. The vertical asymptote(s) is/are x = (Use a comma to separate answers as needed.) There is no vertical asymptote. Find the vertical, horizontal and oblique asymptotes, if any, of the given rational function. R(x) = x3 - 64 / x2 - 6x + 8 Find the vertical asymptote(s), if any, of the function. Select the correct choice below and fill in any answer boxes within your choice. The vertical asymptote(s) is/are x = (Simplify your answer. Use a comma to separate answers as needed.) There is no vertical asymptote. Find the vertical, horizontal and oblique asymptotes, if any, for the given rational function. Q(x) = 2x2 - 7x - 4 / 3x2 - 7x - 20 Select the correct choice below and fill in any answer boxes within your choice. The vertical asymptote(s) is/are x = (Use a comma to separate answers as needed. Use integers or fractions for any numbers in the expression.) There is no vertical asymptote. Analyze the graph of the function. R(x) = x + 14 / x(x + 15) What is the domain of R(x)? {x x 0 andx - 16} {x x 0 andx - 16 andx - 14} {x x 0 and x - 14} All real numbers Analyze the graph of the function. R(x) = 5x + 5 / 4x + 12 What is the domain of R(x)? {x|x 0 and x -3 and x - 1} {x|x -3} {x|x and x - 3} All real numbers Analyze the graph of the function.R(x) = x2 / x2 + x - 56 What is the domain of R(x)? {x|x 7 and x - 8} {x|x 0,x -1, and x 8} {x|x 0, x 7, and x - 8} All real numbers

Show transcribed image text Complete the sentence below. If r is a real zero of even multiplicity of a function f then the graph of f the x-axis at r. If r is a real zero of even multiplicity of a function f then the graph of f the x-axis at r. Complete the sentence below. The graphs of power functions of the form f(x) = x n: where n is an even integer, always contain the points and The graphs of power functions of the form f(x) = x n: where n is an even integer, always contain the points and If r is a solution to the equation f(x) = 05 name three additional statements that can be made about f and r assuming f is a polynomial function. Choose the correct answer below. If r is a solution to the equation f(x) = 0, then r is a real zero of the polynomial function r is an x-intercept of the graph of and x - r is a factor of f. If r is a solution to the equation f(x) = 0, then r is a complex zero of the polynomial function r is an x-intercept of the graph of and x + r is a factor of f. If r is a solution to the equation f(x) = 0, then r is a real zero of the polynomial function r is a y-intercept of the graph of x and x + r is a factor of f. Explain what the notation Km f(x) = -infinity means. x rightarrow infinity Choose the correct answer below. The limit of f(x) as x approaches - infinity equals infinity. The limit of x as f(x) approaches infinity equals - infinity. The limit of f(x) as x approaches infinity equals - infinity. The limit of x as f(x) approaches - infinity equals infinity. Form a polynomial whose zeros and degree are given. Zeros: -7, multiplicity 1; - 3, multiplicity 2; degree 3 Type a polynomial with integer coefficients and a leading coefficient of 1 in the box below. F(x) = Find the vertical horizontal and oblique asymptotes, if any, for the following rational function.R(x) = 12x / x + 20 Select the correct choice below and fill in any answer boxes within your choice. The vertical asymptote(s) is/are x = (Use a comma to separate answers as needed.) There is no vertical asymptote. Find the vertical, horizontal and oblique asymptotes, if any, of the given rational function. R(x) = x3 - 64 / x2 - 6x + 8 Find the vertical asymptote(s), if any, of the function. Select the correct choice below and fill in any answer boxes within your choice. The vertical asymptote(s) is/are x = (Simplify your answer. Use a comma to separate answers as needed.) There is no vertical asymptote. Find the vertical, horizontal and oblique asymptotes, if any, for the given rational function. Q(x) = 2x2 - 7x - 4 / 3x2 - 7x - 20 Select the correct choice below and fill in any answer boxes within your choice. The vertical asymptote(s) is/are x = (Use a comma to separate answers as needed. Use integers or fractions for any numbers in the expression.) There is no vertical asymptote. Analyze the graph of the function. R(x) = x + 14 / x(x + 15) What is the domain of R(x)? {x x 0 andx - 16} {x x 0 andx - 16 andx - 14} {x x 0 and x - 14} All real numbers Analyze the graph of the function. R(x) = 5x + 5 / 4x + 12 What is the domain of R(x)? {x|x 0 and x -3 and x - 1} {x|x -3} {x|x and x - 3} All real numbers Analyze the graph of the function.R(x) = x2 / x2 + x - 56 What is the domain of R(x)? {x|x 7 and x - 8} {x|x 0,x -1, and x 8} {x|x 0, x 7, and x - 8} All real numbers

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Algebra

Which rational function has the given graph? R(x) = 2x2/x2-4 R(x) = 2x2/x2+4 R(x) = -2x2/x2+4 R(x) = 2x2/x2+4 (R(x) = -2x2/x2-4 Use a transformation of the graph of y = x5 to graph the function. h(x) = -4x5 Choose the correct graph below. Determine whether the function is a polynomial function. If it is, state the degree. If it is not tell why not. g(x)= 1-x3/5 Choose the correct answer below. Polynomial of degree 3 Not a polynomial because of the negative coefficient of x3 Polynomial of degree 1 Not a polynomial because of the negative power of x Find the vertical, horizontal and oblique asymptotes, if any, of the given rational function. R(x) = x3 - 27/x2-x-6 Find the vertical asymptote(s), it any, of the function. Select the correct choice below and fill in any answer boxes within your choice. The vertical asymptote(s) is/are x = . (Simplify your answer. Use a comma to separate answers as needed.) There is no vertical asymptote. Find the horizontal asymptote(s), if any, of the function. Select the correct choice below and fill in any answer boxes within your choice. The horizontal asymptote(s) is/are y = . (Simplify your answer. Use a comma to separate answers as needed.) There is no horizontal asymptote. Find the oblique asymptote(s), if any, of the function. Select the correct choice below and fill in any answer boxes within your choice. The oblique asymptote(s) is/are y = . (Simplify your answer. Use a comma to separate answers as needed.) . There is no oblique asymptote. Use a transformation of the graph of y= x4 to graph the function. f(x) = (x+7)4 Find the vertical, horizontal and oblique asymptotes, if any, for the following rational function. R(x) = 8x/x+16 Select the correct choice below and till m any answer boxes within your choice. The vertical asymptote(s) is/are x = . (Use a comma to separate answers as needed.) There is no vertical asymptote. Select the correct choice below and fill in any answer boxes within your choice. The horizontal asymptote(s) is/are y = . (Use a comma to separate answers as needed.) There is no horizontal asymptote. Select the correct choice below and fill in any answer boxes within your choice. The oblique asymptote(s) is/are y = . (Use a comma to separate answers as needed.) There is no oblique asymptote. Which rational function could have the given graph? In the functions shown, c is a constant. Choose the correct answer below. R(x) = (x+1)2(x+4)(x+c)/(x-1)(x+3) R(x) = (x+1)(x+4)(x2+c)/(x-1)2(x+3)2 R(x) = (x+1)(x+4)2(x+c)/(x-1)(x+3) R(x) = (x+1)(x+4)(x2+c)/(x-1)2(x+3)2 Analyze the graph of the function. R(x) = x(x-9)2/(x+12)3 What is the domain of R(x)? {x|x 0, x 9, and x - 12} {x|x -12} {x|x 0 and x - 12} All real numbers What is the equation of the vertical asymptote(s) of R(x)? Select the correct choice below and fill in any answer boxes within your choice. x= (Use a comma to separate answers as needed. Type an integer or a fraction.) There is no vertical asymptote. What is the equation of the horizontal or oblique asymptote of R(x)? Select the correct choice below and fill in any answer boxes within your choice. y = (Simplify your answer.) There is no horizontal or oblique asymptote. Choose the correct graph for R(x) below. Find the vertical horizontal and oblique asymptotes, if any, for the following rational function. T(x) = x3/x4 - 1 Select the correct choice below and fill in any answer boxes within your choice. The vertical asymptote(s) is/are x = . (Use a comma to separate answers as needed.) There is no vertical asymptote. Select the correct choice below and fill in any answer boxes within your choice. The horizontal asymptote(s) is/are y = . (Use a comma to separate answers as needed.) There is no horizontal asymptote. Select the correct choice below and fill in any answer boxes within your choice. The oblique asymptote(s) is/are y= . (Use a comma to separate answers as needed.) There is no oblique asymptote.

Show transcribed image text Which rational function has the given graph? R(x) = 2x2/x2-4 R(x) = 2x2/x2+4 R(x) = -2x2/x2+4 R(x) = 2x2/x2+4 (R(x) = -2x2/x2-4 Use a transformation of the graph of y = x5 to graph the function. h(x) = -4x5 Choose the correct graph below. Determine whether the function is a polynomial function. If it is, state the degree. If it is not tell why not. g(x)= 1-x3/5 Choose the correct answer below. Polynomial of degree 3 Not a polynomial because of the negative coefficient of x3 Polynomial of degree 1 Not a polynomial because of the negative power of x Find the vertical, horizontal and oblique asymptotes, if any, of the given rational function. R(x) = x3 - 27/x2-x-6 Find the vertical asymptote(s), it any, of the function. Select the correct choice below and fill in any answer boxes within your choice. The vertical asymptote(s) is/are x = . (Simplify your answer. Use a comma to separate answers as needed.) There is no vertical asymptote. Find the horizontal asymptote(s), if any, of the function. Select the correct choice below and fill in any answer boxes within your choice. The horizontal asymptote(s) is/are y = . (Simplify your answer. Use a comma to separate answers as needed.) There is no horizontal asymptote. Find the oblique asymptote(s), if any, of the function. Select the correct choice below and fill in any answer boxes within your choice. The oblique asymptote(s) is/are y = . (Simplify your answer. Use a comma to separate answers as needed.) . There is no oblique asymptote. Use a transformation of the graph of y= x4 to graph the function. f(x) = (x+7)4 Find the vertical, horizontal and oblique asymptotes, if any, for the following rational function. R(x) = 8x/x+16 Select the correct choice below and till m any answer boxes within your choice. The vertical asymptote(s) is/are x = . (Use a comma to separate answers as needed.) There is no vertical asymptote. Select the correct choice below and fill in any answer boxes within your choice. The horizontal asymptote(s) is/are y = . (Use a comma to separate answers as needed.) There is no horizontal asymptote. Select the correct choice below and fill in any answer boxes within your choice. The oblique asymptote(s) is/are y = . (Use a comma to separate answers as needed.) There is no oblique asymptote. Which rational function could have the given graph? In the functions shown, c is a constant. Choose the correct answer below. R(x) = (x+1)2(x+4)(x+c)/(x-1)(x+3) R(x) = (x+1)(x+4)(x2+c)/(x-1)2(x+3)2 R(x) = (x+1)(x+4)2(x+c)/(x-1)(x+3) R(x) = (x+1)(x+4)(x2+c)/(x-1)2(x+3)2 Analyze the graph of the function. R(x) = x(x-9)2/(x+12)3 What is the domain of R(x)? {x|x 0, x 9, and x - 12} {x|x -12} {x|x 0 and x - 12} All real numbers What is the equation of the vertical asymptote(s) of R(x)? Select the correct choice below and fill in any answer boxes within your choice. x= (Use a comma to separate answers as needed. Type an integer or a fraction.) There is no vertical asymptote. What is the equation of the horizontal or oblique asymptote of R(x)? Select the correct choice below and fill in any answer boxes within your choice. y = (Simplify your answer.) There is no horizontal or oblique asymptote. Choose the correct graph for R(x) below. Find the vertical horizontal and oblique asymptotes, if any, for the following rational function. T(x) = x3/x4 - 1 Select the correct choice below and fill in any answer boxes within your choice. The vertical asymptote(s) is/are x = . (Use a comma to separate answers as needed.) There is no vertical asymptote. Select the correct choice below and fill in any answer boxes within your choice. The horizontal asymptote(s) is/are y = . (Use a comma to separate answers as needed.) There is no horizontal asymptote. Select the correct choice below and fill in any answer boxes within your choice. The oblique asymptote(s) is/are y= . (Use a comma to separate answers as needed.) There is no oblique asymptote.

Algebra

Which rational function could have the given graph? In the functions shown, c is a constant. Choose the correct answer below. Use a transformation of the graph of y = x4 to graph the function. f(x) = (x + 8)4 Select the graph of f(x) = (x + 8)4. Find the vertical, horizontal, and oblique asymptotes, it any. or the given rational function. Find the vertical asymptote(s), if any, of the function. Select the correct choice below and fill in any answer boxes within your choice. The vertical asymptote(s) is/are x = . (Simplify your answer. Use a comma to separate answers as needed.) There is no vertical asymptote. Find the horizontal asymptote(s), if any, of the function. Select the correct choice below and fill in any answer boxes within your choice. The horizontal asymptote(s) is/are y = . (Simplify your answer. Use a comma to separate answers as needed.) There is no horizontal asymptote. Find the oblique asymptote(s), if any, of the function. Select the correct choice below and fill in any answer boxes within your choice. The oblique asymptote(s) is/are y = . (Simplify your answer. Use a comma to separate answers as needed.) There is no oblique asymptote. Find the vertical, horizontal, and oblique asymptotes, if any, for the following rational function. Select the correct choice below and fill in any answer boxes within your choice. The vertical asymptote(s) is/are x = . (Use a comma to separate answers as needed.) There is no vertical asymptote. Select the correct choice below and fill in any answer boxes within your choice. The horizontal asymptote(s) is/are y = . (Use a comma to separate answers as needed.) There is no horizontal asymptote. Select the correct choice below and fill in any answer boxes within your choice. The oblique asymptote(s) is/are y = . (Use a comma to separate answers as needed.) There is no oblique asymptote. Which rational function has the given graph? Find the vertical, horizontal, and oblique asymptotes, if any. for the following rational function. Select the correct choice below and fill in any answer boxes within your choice. The vertical asymptote(s) is/are x = . (Use a comma to separate answers as needed.) There is no vertical asymptote. Select the correct choice below and fill in any answer boxes within your choice. The horizontal asymptote(s) is/are y = (Use a comma to separate answers as needed.) There is no horizontal asymptote. Select the correct choice below and fill in any answer boxes within your choice. The oblique asymptote(s) is/are y = . (Use a comma to separate answers as needed.) There is no oblique asymptote. Use a transformation of the graph of y = x5 to graph the function. h(x) = -4x5 Choose the correct graph below. Analyze the graph or the function. What is the domain of R(x)? {x|x - 14} {x|x 0. x 17, and x - 14} {x|x 0 and x - 14} All real numbers What is the equation of the vertical asymptote(s) of R(x)? Select the correct choice below and fill in any answer boxes within your choice. x - (Use a comma to separate answers as needed. Type an integer or a fraction.) There is no vertical asymptote. What is the equation of the horizontal or oblique asymptote of R(x)? Select the correct choice below and fill in any answer boxes within your choice. y = (Simplify your answer.) There is no horizontal or oblique asymptote. Choose the correct graph for R(x) below. Determine whether the function is a polynomial function. If it is, state the degree. If it is not, tell why not. Choose the correct answer below. Polynomial of degree 6 Polynomial of degree 3 Not a polynomial because of the negative coefficient of x3 Not a polynomial because of the negative power of x

Show transcribed image text Which rational function could have the given graph? In the functions shown, c is a constant. Choose the correct answer below. Use a transformation of the graph of y = x4 to graph the function. f(x) = (x + 8)4 Select the graph of f(x) = (x + 8)4. Find the vertical, horizontal, and oblique asymptotes, it any. or the given rational function. Find the vertical asymptote(s), if any, of the function. Select the correct choice below and fill in any answer boxes within your choice. The vertical asymptote(s) is/are x = . (Simplify your answer. Use a comma to separate answers as needed.) There is no vertical asymptote. Find the horizontal asymptote(s), if any, of the function. Select the correct choice below and fill in any answer boxes within your choice. The horizontal asymptote(s) is/are y = . (Simplify your answer. Use a comma to separate answers as needed.) There is no horizontal asymptote. Find the oblique asymptote(s), if any, of the function. Select the correct choice below and fill in any answer boxes within your choice. The oblique asymptote(s) is/are y = . (Simplify your answer. Use a comma to separate answers as needed.) There is no oblique asymptote. Find the vertical, horizontal, and oblique asymptotes, if any, for the following rational function. Select the correct choice below and fill in any answer boxes within your choice. The vertical asymptote(s) is/are x = . (Use a comma to separate answers as needed.) There is no vertical asymptote. Select the correct choice below and fill in any answer boxes within your choice. The horizontal asymptote(s) is/are y = . (Use a comma to separate answers as needed.) There is no horizontal asymptote. Select the correct choice below and fill in any answer boxes within your choice. The oblique asymptote(s) is/are y = . (Use a comma to separate answers as needed.) There is no oblique asymptote. Which rational function has the given graph? Find the vertical, horizontal, and oblique asymptotes, if any. for the following rational function. Select the correct choice below and fill in any answer boxes within your choice. The vertical asymptote(s) is/are x = . (Use a comma to separate answers as needed.) There is no vertical asymptote. Select the correct choice below and fill in any answer boxes within your choice. The horizontal asymptote(s) is/are y = (Use a comma to separate answers as needed.) There is no horizontal asymptote. Select the correct choice below and fill in any answer boxes within your choice. The oblique asymptote(s) is/are y = . (Use a comma to separate answers as needed.) There is no oblique asymptote. Use a transformation of the graph of y = x5 to graph the function. h(x) = -4x5 Choose the correct graph below. Analyze the graph or the function. What is the domain of R(x)? {x|x - 14} {x|x 0. x 17, and x - 14} {x|x 0 and x - 14} All real numbers What is the equation of the vertical asymptote(s) of R(x)? Select the correct choice below and fill in any answer boxes within your choice. x - (Use a comma to separate answers as needed. Type an integer or a fraction.) There is no vertical asymptote. What is the equation of the horizontal or oblique asymptote of R(x)? Select the correct choice below and fill in any answer boxes within your choice. y = (Simplify your answer.) There is no horizontal or oblique asymptote. Choose the correct graph for R(x) below. Determine whether the function is a polynomial function. If it is, state the degree. If it is not, tell why not. Choose the correct answer below. Polynomial of degree 6 Polynomial of degree 3 Not a polynomial because of the negative coefficient of x3 Not a polynomial because of the negative power of x

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