Ch01+Linear+Relations

Chapter One - Linear Relations and Functions
Chapter One objectives, topics, and assignments:

Chapter One highlights in Powerpoint:

And here they are as a PDF file:

1-1 Relations and Functions

 * A relation is a set of ordered pairs.
 * Ex: V(x) = {(peas, carrots), (macaroni, cheese), (hamburger, fries), (hamburger, onion rings) }.
 * Ex: q(x) = 7x + 4
 * All functions are relations, but not all relations are functions. How can you tell the difference?
 * Here is a hint: y = x 2 is a function. However, y = +/- x (1/2) is not.
 * The **//domain//** of a relation is the set of all values for which the relation is defined (possible inputs).
 * Domain of V(x) = { peas, macaroni, hamburger } (Hamburger should be listed only once.)
 * The //**range**// of a relation is the set of all possible values that the relation can generate (possible outputs).
 * Range of q(x) = { all real values }
 * You can imagine a relation (or function) as a machine that processes inputs and converts them to outputs.

1-2 Composition and Inverses of Functions
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 * Functions can be added, subtracted, multiplied, and divided to generate new functions. These new functions are called **compositions**.
 * Question: If you create a composition of two functions, will the result always be a function, or is it possible that the composition might //not// be a function?
 * Compositions also can be combined in the following way: use the //output// of one function as the //input// of another function. Example: //f(x)// = 3x+1, //g(x)// = x 2 . So //g( f(x) )// is g( 3x+1 ) = ( 3x + 1 ) 2 = 9x 2 + 6x + 1.
 * In words we would pronounce //g( f(x) )// as "GEE of EFF of EX."
 * What about //f( g(x) )//? Is this the same as //g( f(x) )//? Maybe. See if it is for //f(x)// and //g(x)// as defined above.
 * Our textbook uses the notation [//f o g//] to mean the same a //f( g(x) )//. See the yellow box at the bottom of Page 13 in the textbook.
 * The domain of //f o g// includes all of the elements //x// in the domain of //g// for which //g(x)// is in the domain of //f//.
 * One should be careful to identify the domain of a composition. For instance, suppose h(x) = f(x) / g(x). Separately f(0) and g(0) both are defined: f(0) = 1, and g(0) = 0. But h(0) has a problem. Why?


 * The inverse of a function or relation is the operation that "undoes" whatever the original function or relation did.
 * Ex: Suppose //p(x)// = 4x + 3. To "undo" this one first must subtract 3, then divide the result by 4.
 * Therefore, the inverse of //p(x)// is //p –1 (x)// = ( x - 3 ) / 4.
 * It is easy to be confused by the notation. //p –1 (x)// is NOT the reciprocal of //p(x)//.
 * One other point: Some functions have inverses that are also functions, but some functions do not. Why? And how can you tell //graphically// ahead of time whether a function's inverse also is a function?
 * Finally, can you think of a function that is its own inverse? See if you can identify a function //f//(//x//) such that :
 * //f// –1 (//f//(//x//)) = //x//

1-4 Distance and Slope

 * The distance formula is merely a modified statement of the Pythagorean Theorem:
 * d = sqrt[(x1–x2)^2 + (y1–y2)^2].
 * Slope formula: m = (y2–y1) / (x2–x1) -- also stated as "rise over run"
 * The midpoint of a segment whose endpoints are (x1, y1) and (x2, y2) is simply the "average" of the two endpoints. Just take the average of the x's to obtain the x-coordinate of the midpoint; and take the average of the y's to get the y-coordinate of the midpoint.

1-5 Forms of Linear Equations

 * Slope-intercept: y = mx+b
 * Standard: Ax +By +C = 0 {Some folks write this as Ax + By = C, but then C takes on the opposite sense.}
 * Point-slope: y – y1 = m(x – x1) --- This is a generalized restatement of the slope formula.

1-6 Parallel and Perpendicular Lines

 * Parallel lines have the same slope.
 * Ex: y = 3x + 5, y = 3x - 7: same slope, so these two lines are parallel.
 * Ex: 5x – 3y + 8 = 0, 5x – 3y + 15 = 0: same coefficients for //x// and //y//, so these lines are parallel.
 * Perpendicular lines have slopes that are the negative reciprocal of each other.
 * E: y3 = 4x + 8, y4 = -(1/4) +13: 4 and -(1/4) are negative reciprocals, so these lines are perpendicular.

Chapter One test prep

 * As we approach the end of Chapter One I will supply a set of sample problems to help you prepare for the chapter test in hard copy (Done: handed out in class Sep 9th).


 * Sometime after the test prep has been distributed, I will post a solution to all the problems here. Stay tuned!
 * And here it is! [[file:Ch01 Test Prep solution.pdf]]