Chapter 4 and half
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- A rational function is a function that can be written as the ratio of two polynomial functions. Graphs of rational functions may have breaks in continuity where there is a vertical asymptote or point of discontinuity. - Mathematician Maria Gaetana Agnesi discussed the "curve of Agnesi", the equation x2/y = a2(a - y), which can be written as a rational function y = a2/x2 + a2. - A polynomial is the sum of monomial terms. The degree of a polynomial is the degree of the highest-degree term. The FOIL method can be used to multiply binomials by distributing and combining like terms.
Chapter 4 and half
- 1. Rational Functions A rational function is of the form 푓(푥) = 푝 (푥) , where p(x) and q(x)are polynomial functions 푞 (푥) and 푞(푥) ≠ 0. A graphing calculator is a good tool for exploring graphs of rational functions. Graphs of rational functions may have breaks in continuity. This means that, unlike polynomials functions which can be traced with a pencil that never leaves the paper, a rational function may not be traceable. Breaks in continuity can occur where there is a vertical asymptote or point discontinuity. Point of discontinuity is like a hole in the graph. Vertical asymptote and point of discontinuity occur for the values of x that make the denominator of the rational function zero. Graphing Rational Functions Connection: Mathematical History Mathematician Maria Gaetana Agnesi was one of the greatest women scholars of all time. In the analytic geometry section of her book Analytical Institutions, Agnesi discussed the characteristics of the equation 푥 2푦 = 푎2(푎 − 푦), called the “curve of Agnesi”. The equation can be expressed as 푦 = 푎2 푥2 +푎2. Because the function described above is the ratio of two polynomial expression a3 and 푥 2 + 푎2 is called a rational function. A rational function is function of the form 푓 (푥) = 푝(푥) 푞(푥) , where p(x) and q(x) are polynomial functions and q(x)≠ 0 Examples of Rational Function: 풇(풙) = 푥 푥−1 푔(푥) = 3 푥−3 ℎ(푥) = 푥+1 (푥+2)(푥−5) The lines that graph of the rational function approaches is called Asymptote. If the function is not define when 푥 = 푎, then either there is a “hole” in the graph 푥 = 푎.
- 2. POLYNOMIALS The expression x2+2xy+y2 is called a polynomial. A polynomial is a monomial or a sum of monomials. The monomials that make up the polynomial are called the terms of the polynomial. The two monomials xy and xy ca be combined because they are like terms. Like terms are two monomials that are the same, or differ only by their numerical coefficient. An expression like m2+7mb+12cd with three unliked terms is called trinomial. An expression like xy+b3 with two unliked terms is called binomials. The degree of a polynomial is the degree of the monomial with the greatest degree. Thus, the degree of x2+2xy+y2 is 2. Example: Determine whether or not each expression is a polynomial. Then state the degree of each polynomial. a. 2 7 x4y3 – x3 This expression is a polynomial. The degree of the first term is 4 + 3 or 7, and the degree of the second term is 3. The degree of the polynomial is 7. b. 9 + √푥 − 3 This expression is not polynomials because √푥 is not a monomial. The FOIL Method is an application of the distributive property that make the multiplication easier. FOIL Method of Multiplying Polynomial The product of two binomial is he sum of the products of: F the first terms O the outer terms I the inner terms L the last terms
- 3. Example: Find (k2 +3k +9) (k +3) (k2 +3k +9) (k +3) = 푘2 (k+3) + 3k (k +3) +9(k +3) distributive property =k2∙k+k2∙ 3 + 3푘 ∙ +9 ∙ 푘 + 3 ∙ 9 distributive property = 푘2∙ 푘 + 푘2∙ 3푘2+9푘 + 9푘 + 27 =k2+6푘2+18푘 + 27 combined like terms Dividing Polynomials You can use a process similar to long division of a whole numbers to divide a polynomial by a polynomial when doing the division, remember that you can only add ad subtract like terms. Example: Simplify: c2 –c –30 c –6 c 푐 − 6√푐2 − 푐 − 30 푐 2−6 5푐−30 −푐 − (−6푐) = −푐 + 6푐 표푟 5푐 푐 + 5 푐 − 6√푐2 − 푐 − 30 푐 2−6 5푐−30 5푐−30 0
- 4. Polynomial functions A polynomial function is a function that can be defined by evaluating a polynomial. A function f of one argument is called a polynomial function if it satisfies for all arguments x, where n is a non-negative integer and a0, a1, a2, ..., an are constant coefficients. For example, the function f, taking real numbers to real numbers, defined by is a polynomial function of one variable. Polynomial functions of multiple variables can also be defined, using polynomials in multiple indeterminates, as in An example is also the function which, although it doesn't look like a polynomial, is a polynomial function on since for every from it is true that Polynomial functions are a class of functions having many important properties. They are all continuous, smooth, entire, computable, etc Graphs of polynomial function A polynomial function in one real variable can be represented by a graph. The graph of the zero polynomial f(x) = 0 is the x-axis. The graph of a degree 0 polynomial f(x) = a0, where a0 ≠ 0, is a horizontal line with y-intercept a0 The graph of a degree 1 polynomial (or linear function)
- 5. f(x) = a0 + a1x , where a1 ≠ 0, is an oblique line with y-intercept a0 and slope a1. The graph of a degree 2 polynomial f(x) = a0 + a1x + a2x2, where a2 ≠ 0 is a parabola. The graph of a degree 3 polynomial f(x) = a0 + a1x + a2x2, + a3x3, where a3 ≠ 0 is a cubic curve. The graph of any polynomial with degree 2 or greater f(x) = a0 + a1x + a2x2 + ... + anxn , where an ≠ 0 and n ≥ 2 is a continuous non-linear curve. The graph of a non-constant (univariate) polynomial always tends to infinity when the variable increases indefinitely (in absolute value) Polynomial graphs are analyzed in calculus using intercepts, slopes, concavity, and end behavior. Polynomial of degree 2: f(x) = x2 − x − 2 = (x + 1)(x − 2)
- 6. Polynomial of degree 3: f(x) = x3/4 + 3x2/4 − 3x/2 − 2 = 1/4 (x + 4)(x + 1)(x − 2) Polynomial of degree 4: f(x) = 1/14 (x + 4)(x + 1)(x − 1)(x − 3) + 0.5 Polynomial of degree 5: f(x) = 1/20 (x + 4)(x + 2)(x + 1 )(x − 1)(x − 3) + 2
- 7. Polynomial of degree 2: f(x) = x2 − x − 2 = (x + 1)(x − 2) Polynomial of degree 3: f(x) = x3/4 + 3x2/4 − 3x/2 − 2 = 1/4 (x + 4)(x + 1)(x − 2) Polynomial of degree 4: f(x) = 1/14 (x + 4)(x + 1)(x − 1)(x − 3) + 0.5
- 8. Polynomial of degree 5: f(x) = 1/20 (x + 4)(x + 2)(x + 1 )(x − 1)(x − 3) + 2 Inverse function Definition of Inverse Function Two function f and g are inverse function if and only if both of their compositions are the identity function. That is, (푓 ∘ 푔) = 푥 and (푔 ∘ 푓)(푥) = 푥 An inverse function is a function that "reverses" another function: if the function f applied to an input x gives a result of y, then applying its inverse function g to y gives the result x, and vice versa. i.e., f(x) = y if and only if g(y) = x. A function f that has an inverse is said to be invertible. When it exists, the inverse function is uniquely determined by f and is denoted by f −1, read f inverse. Superscripted "−1" does not, in general, refer to numerical exponentiation. In some situations, for instance when f is an invertible real-valued function of a real variable, the relationship between f andf−1 can be written more compactly, in this case, f−1(f(x)) = x = f(f−1(x)), meaning f−1 composed with f, in either order, is the identity function on R. Property of Inverse Function Suppose 푓 and푓−1 are inverse function. Then 푓(푎) = 푏 and only if 푓−1(푏) = 푎
- 9. Definition of inverse Relationship Two relationships are inverse relationship if and only if whenever one relation contains the element ( a, b ), the other relation contains the element ( b, a )