Exponents are simply a way to abbreviate writing out a long mul ... - PDF document

Exponents are simply a way to abbreviate writing out a long mul Ope Exponents and Radicals: Practice Exercises Simplify the following as much as poss Thus the definition of a logarithm is: logb A = n if and only if bn = A. The base used for logs can vary, but the base is always positive. This means that the "argument", or A, is always positive. (Why?) The most commonly used bases are 10 and the natural logarithm, whose base is e at the base is not written in this special logarithm) and translates to 10n = A. The natural logarithm is written lnA= n (this is the convention for abbreviating loge A= n) and translates to en = A . Good t 8. log5 N = -2 9. log100 N = log 2= 0.301 log 3= Ñ log and so on. Now try a different equation, this time involving multiplication instead of addition: a¥b=0. How many solutions are there to this problem? Only two: either a=0 or b=0. For ax + ay = a(x + x2 +(a+b)x + 4. x4 Ð a4 = (x2 Ð a2 )(x2 +a2 ) x4 Ð16 = (x2 Ð4)(x2 +4) = (xÐa)(x+a)(x2 +a2 ) = (xÐ2)(x+2)(x2 +4) Sum or Difference of Cube Ð a3 = (xÐa)(x2 + ax+a2 ) x3 Ð27 = (xÐ3)(x2 +3x+9) Factoring by G x2+2xÐ4x+6 , and 1x2 +1 . Algebraic fractions that have a polynomial in both the num 5%&'()*xx = 10+3x5x 2. Subtract fractions (find a commo 3x3y = xy ab+acad = a(b+c)ad = b+c 6. Rationalizing a. If the numerator or de c. If the numerator or denominator is a +b , multiply by a Ñba Ñb Algebraic Fractions: Practice Exercises Simpify as com solutions or roots. The solutions of an equation are not affected as long as you add,subtract, multiply or divide both sides of the equation by the same quantity. The only exception is dividing by zero. The method of solving an equation depends on the degree of the equation. First degree equations are solved using addition, subtraction, multiplication and division. ex. 3x + 7 = 9 3x = 2 Subtract 7 from both sides. 3x3 = 23 Divid The symbol is read: � greater than less than # greater than OR equal to " less than OR equal to An important distinction to remember about algebraic inequalities is that, generally, they have an infinite number of real solutions. An algebraic equation has one, two or more finite real solutions. SOLVING LINEAR INEQUALITIES Linear inequalities are solved in a similar manner to solving linear equations, with one important difference. Like equations, any value added or subtracted from one side of the inequality must also be equivalently added or subtracted from the other side. The same is true for multiplying and dividing both sides by posititve numbers. However, multiplying or dividing both sides Ð 10 20 + x Ð x 30 + x add10 to both sides Ð 2x 30 subtract x Ex 3. x2 Ð 3x Ð 10 " 0 (x Ð 5)(x + 2) " 0 factored form x = Ð2,5 roots (where each term = 0) Place these roots on a number line, us This number line method works well for any higher power inequality with many factored terms or rational (division) inequalities. Caution: Problems which are rational (like Practice exercises 7 and 8) may have both open and closed circles. Even if you have and inequality which includes "equals" (# or "), you must never use a closed circle on a root which causes you to divi 4(x Ð 3)4 0 14. (x Ð 5)(x + 2)7 Ð (x handout "Graphs You Should Know" will help you familiarize yourself with the most common graphs. Lines Ax +By + C = 0 Exponents on x and y are both one. As stated above, you need only to find the y-intercept (let x = 0 and solve for y) and the x-intercept (let y = 0 and solve for x) and draw a line through them. Absolute Values y - k = |x Ð h| Exponents on x and y are both one. Absolute value signs around the x-term. (If they are around the y-term it is "sideways" and not a function.) Once you have found your intercepts you must graph its vertex. Trigonometric y = cosx y = tanx The rules of translating any basic function are as follows: a. y = f(x) + C is moved up C units from y = f(x). b. y = f(x) Ð C is moved down C uni Graph the followi Ð 2 17. y = x 18. x = Ð1 Functions A function is a relationship between two variables where each value of the independent variable only corresponds exactly to one value of the dependent variable . In layman's terms, every x has only one y. Another way to look at it is, if you could list all the ordered pairs, no two unique ordered pairs have the same first component. Graphically, a function passes the "vertical line test"; any vertical line hits the graph only once. Some examples of f f(x) = ex Exponential function g(x) = ln(x) Logarithmic Two functions, f(x) and g(x), are inverses if the components of every order (x) . Be careful! When used in this way with functions this notation never means "reciprocal". It always means "inverse function". To prove two functions, f(x) and g(x), are inverses you must show that for every x in the domain of g(x), f(g(x)) = x, and for every x in the domain of f(x), g(f(x)) = x. Functions: Practice Exercises 1. Let f(x) = a. f(0) a. b. c. d. a. Which of the graphs is a function? b. For the one(s) which are function(s), give the domain and range. 3. Find the domain of each of the given functions: a. f(x) = 4 - x2 b. g(x) = =(42 Ð3x)[(42 )2 + 48x + (3x)2 ] = (16Ð3x)(256 + 48x + 9x2 ) 9. x(x+3) + w(x+3) = (x+w)(x+3) 10. 1Ð(x2 17. 18. Functions 1. a. 0 b. 2 c. 6. a. f