Module 2 Lesson 4 Notes
... What is a parent function? We use the term ‘parent function’ to describe a family of graphs. The parent function gives a graph all of the unique properties and then we use transformations to move the graph around the plane. You have already seen this with lines. The parent function for a lines is y ...
... What is a parent function? We use the term ‘parent function’ to describe a family of graphs. The parent function gives a graph all of the unique properties and then we use transformations to move the graph around the plane. You have already seen this with lines. The parent function for a lines is y ...
2.7_Polynomials Rat Inequalities
... Locating the x-intercepts of a polynomial function, is an important step in finding the solution set for polynomial inequalities in the form f(x) > 0 or f(x) < 0. We use the x-intercepts of f as boundary points that divide the real number line into intervals. On each interval, the graph of f is eith ...
... Locating the x-intercepts of a polynomial function, is an important step in finding the solution set for polynomial inequalities in the form f(x) > 0 or f(x) < 0. We use the x-intercepts of f as boundary points that divide the real number line into intervals. On each interval, the graph of f is eith ...
solution set for the homework problems
... c) From (b), p = 2 − q is an irrational number and 0 < p < d − c from (a). Thus we get c < c + p < d. As seen in (b), c + p cannot be rational. Because if c + p = a is rational, then p = a − c has to be rational which was just proven not to be the case. ...
... c) From (b), p = 2 − q is an irrational number and 0 < p < d − c from (a). Thus we get c < c + p < d. As seen in (b), c + p cannot be rational. Because if c + p = a is rational, then p = a − c has to be rational which was just proven not to be the case. ...
6.2 One-to-One and Inverse Functions
... for each y in the domain of the inverse function there is a unique x in the range, it is a one-to-one function. If a horizontal line intersects the graph of a function f no more than once, then f is one-to-one. Only one-to-one functions have inverses. We can verify that f and f –1 are inverses showi ...
... for each y in the domain of the inverse function there is a unique x in the range, it is a one-to-one function. If a horizontal line intersects the graph of a function f no more than once, then f is one-to-one. Only one-to-one functions have inverses. We can verify that f and f –1 are inverses showi ...
Chapter 2 Algebra Review 2.1 Arithmetic Operations
... We briefly cover an extension of the Perfect Square formula we had, namely (a + b)2 = a2 + 2ab + b2. If we multiply both sides by (a+b) and simplify we get another ‘binomial expansion’: (a + b)3 = a3 + 3a2b + 3ab2 + b3. Before we give a general formula for (a + b)n we define the ‘Factorial Notation’ ...
... We briefly cover an extension of the Perfect Square formula we had, namely (a + b)2 = a2 + 2ab + b2. If we multiply both sides by (a+b) and simplify we get another ‘binomial expansion’: (a + b)3 = a3 + 3a2b + 3ab2 + b3. Before we give a general formula for (a + b)n we define the ‘Factorial Notation’ ...
Comments on predicative logic
... case for a proof by induction on the complexity of formulas that the conditional ¬¬A → A is derivable for every formula A of the language. This follows from well-known results in proof theory, since the connectives of our language are all “negative”: ‘→’, ‘∧’ and ‘∀’. With the stability scheme in pl ...
... case for a proof by induction on the complexity of formulas that the conditional ¬¬A → A is derivable for every formula A of the language. This follows from well-known results in proof theory, since the connectives of our language are all “negative”: ‘→’, ‘∧’ and ‘∀’. With the stability scheme in pl ...