Outline for Math 119-Test 1
Date: Monday, June 1, 2009

The format of test 1 will be open questions (ie: not mult. choice or t/f). There will be about 8-12 questions many of which will have parts. Partial credit may be awarded if the work shown warrants it. You will be required to show all of your work in order to receive full credit for a solution. You may use a calculator (not the TI 89 or TI92). You may not use any notes, books, etc.

Disclaimer: The fact that a certain specific type of problem is not named below does not preclude it from being covered on the test. This document is intended to give you an outline of the most important topics that have been covered.

Section A.1: Real Numbers and their Properties

• you should be able to use the properties of real numbers, negatives and fractions
  
• be able to use set notation and interval notation (and know the difference between these two terms)
• you should know the properties of the absolute value and how it is used to compute distance between points on the real line
  

Section A.2: Exponents and Radicals

• the laws of exponents are listed in this section; you will be expected to know these laws well enough to be able to recognize when and how to use them in computational problems
• be able to eliminate negative exponents in an expression
• you should understand the definition of "nth root" and the difference between when n is even or odd
• you should know how to perform computations involving nth roots, including the properties on p. A15
  
• be able to convert from fractional exponents to nth root notation and vice versa
• know what the directions mean for simplifying expressions (i.e.: integer exponents only, etc)
  

• be able to recognize whether or not an expression is a polynomial and know it's domain
  
• know how to combine "like" terms
• know how to distribute and how to multiply binomials (FOIL) and trinomials
• know the special products on p. A25 and the factoring special polynomial forms on p. A 27 [except "cube of a binomial" and "sum or difference of two cubes"]
• know how to factor out common factors
  
• know how to factor a trinomial by reversing the FOIL method
• know what it means to "factor an expression completely"
  

• be able to identify whether or not an expression is a rational expression
• know how to find the domain of a rational expression
• the cancellation law for fractional expressions and be able to use it to simplify expressions
• be able to add and subtract fractional expressions by first finding the least common denominator
• be able to multiply and divide fractional expressions and then simplify appropriately
• be able to rationalize the numerator or denominator of a fractional expression (may need to use the conjugate)
• be able to simplify complex fractions and the difference quotient
  

• be able to solve equations of all the types covered in this section
• linear equations
• fractional equations*
• quadratic equations (by factoring and quadratic formula;  know when it is okay to extract square roots)
• higher degree polynomial equations (by factoring, by grouping, by beginning with graphing calculator)
• equations involving radicals*
• equations with absolute values*
• where appropriate, you may be required to solve equations exactly (no calculator approximations)
  

• know the Properties of Inequalities (p. A61)
• be able to solve linear inequalities, including double inequalities and absolute value inequalities
  

• be very familiar with errors to avoid, as described in this section
• be able to do hard algebraic computations, like those explained and practiced in this section
  

• know what the Cartesian (or coordinate) plane is and all relevant terms (i.e.: quadrants, origin, x-axis, etc.)
• be able to plot points in the coordinate plane; be able to sketch an appropriate, well-labeled scatter plot
  
•   memorize the Pythagorean Theorem, the distance and midpoint formulas and be able to use them

• be able to write solutions to equations in two variables as points in the plane; understand the relationship between points in the plane and solutions to equations in two variables
• know how to sketch the graph of an equation by plotting points (and when this is an appropriate technique)
• know how to compute x and y-intercepts algebraically and what they represent graphically
  
• know symmetric w.r.t the y-axis: graphically, algebraically
• know symmetric w.r.t the x-axis: graphically, algebraically
  
• know symmetric w.r.t to the origin: graphically, algebraically
• memorize the formula for the equation of a circle; be able to find the equation of a circle and sketch its graph; be able to identify a circle's center and radius from its equation

• be able to compute the slope and y-intercept of a line and understand their graphical interpretations (i.e.: slope represents the steepness of a line (how?) and the y-intercept is the value of y where the graph crosses the y-axis (what is the x-value there?))
• be able to find the equation of a line given two points or a point and the slope, or the y-intercept and the slope
• be able to find the equations of parallel and perpendicular lines (how are the slopes related?)
• be able to find the equations of vertical and horizontal lines (what are their slopes?)

• know the definition of function (p. 40) and all related terminology
  
• understand four notions of functions: algebraic, graphical, verbal and numerical
  
• know the phrase "y as a function of x" and similar phrases, what they mean and how to use
• know how to determine whether a function is being expressed(in each of the four forms)
• completely understand the notation used for functions and how to compute using it
• understand piecewise defined functions and be able to compute function values
• know how to find the domain of a function
• be able to compute and simplify difference quotients
  

• know the definition of the graph of a function; be able to use the vertical line test to decide if your graph is the graph of a function
• know how to plot points on the graph of a function and how to read function values from a graph
• be able to find domain and range of a function from its graph
• be able to find the zeros of a function and understand graphically what they represent
• be able to determine on which intervals a function is increasing, decreasing or constant and be able to find relative minimums and maximums from its graph
• memorize formula for average rate of change and be able to use appropriately
  
• be able to read a graph (given a graph, can you answer questions about it such as: find all numbers, x, such that g(x)=3, etc)

• you should be able to identify the parent function of a given function and have a working knowledge of the properties of each parent function discussed in this section, for example:
  
• domain/range
• symmetry (even/odd)
  
• intercepts
• increasing/decreasing
• you should be able to sketch the graph of any parent function and functions that are defined piecewise
  

Section 1.7: Transformations of Functions

• a detailed outline for this section is provided in the handout for the section, including the specific types of problems to master
• you should be able to apply the methods of this section to every type of function that follows (algebraic, exponential and log, trig and inverse trig...)

• know how the rules: f+g, f-g, fg, f/g, are defined; be able to find the formulas for these functions and their domains
• know how the rule for function composition is defined (and the appropriate notation); be able to find rule and domain
• be able to compute the value of these new rules at particular input values (ie: f(g(1)) etc) using algebraic expressions, graphs or tables for f and g
• be able to decompose a function into its composite pieces

• use the handout as a guide; be able to complete the following types of problems:
• determine whether or not a function is one-to-one graphical (either given a graph or using your graphing calculators and knowledge of complete graphs)
• be able to decide if two functions are inverses using the cancellation properties
• be comfortable with the notation, f^(-1)  (which does NOT equal 1/f)
• be able to find the domain and range of the inverse function by using domain and range of the original function (which you may be asked to find)
• be able to find the inverse of a given function f(x);  you may need to restrict the domain to make the function one-to-one
• be able to sketch the graph of the inverse function of f, given either the graph of f or the rule (and using your calculator to find the graph of f)

Good Luck!