MA123, Chapter 1: Equations, functions and graphs (pp. 1-15)

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1 MA123, Chapter 1: Equations, functions and graphs (pp. 1-15) Date: Chapter Goals: Identif solutions to an equation. Solve an equation for one variable in terms of another. What is a function? Understand function notation. Find inverse functions. What is a graph? Sketch the graph of an equation in two variables. Find the equation of a line and a parabola. Find the intersection point(s) of two graphs. Assignments: Assignment 00 Assignment 01 Equations and solution(s) to equations: One wa in which humanit increases its understanding of the universe is b discovering relationships between various objects, concepts, quantities, and so on. Our understanding of a relationship between two quantities is sharpest when this relationship can be completel quantified and epressed in an equation. Roughl speaking, an equation is a statement that two mathematical epressions are equal. For instance, = 5 is an equation relating and. A set of numbers that can be substituted for the variables in an equation so that the equalit is true is a solution for the equation. A solution is said to satisf the equation. Eample 1: Is = 1 and = 2 a solution for the equation 2 + = 2? What about = 1 and = 1? Man problems in the sciences, economics, finance, medicine and numerous other fields can be formulated into algebraic terms b identifing variables epressing unknown quantities and b setting up appropriate equations relating these variables. Eample 2: Suppose a fuel miture is 4% ethanol and 96% gasoline. How much ethanol (in gallons) must ou add to one gallon of fuel so that the new fuel miture is 10% ethanol? 1

2 Equations into functions: An equation in two (or more) variables can sometimes be solved in terms of one of the variables. This tpe of equation is closel related to the notion of a function. Eample 3: Solve the equation = 7 for in terms of. Observe that in the equation = 7 3, the epression on the right-hand side can be viewed as a recipe that associates to an given value of precisel one corresponding value for. Definition of function: A function f is a rule that assigns to each element in a set A eactl one element, called f(), in a set B. The set A is called the domain of f whereas the set B is called the codomain of f; f() is called the value of f at, or the image of under f. The range of f is the set of all possible values of f() as varies throughout the domain: range of f = {f() A}. input f Machine diagram of f f() output A a Arrow diagram of f b f f() B f(a) = f(b) range of f Evaluating a function: The smbol that represents an arbitrar number in the domain of a function f is called an independent variable. The smbol that represents a number in the range of f is called a dependent variable. In the definition of a function the independent variable plas the role of a placeholder. For eample, the function f() = can be thought of as f( ) = To evaluate f at a number (epression), we substitute the number (epression) for the placeholder. Note: If f is a function of, then = f() is a special kind of equation, in which the variable appears alone on the left side of the equal sign and the epression on the right side of the equal sign involves onl the other variable. Conversel, when we have this special kind of equation, such as = e , it is common to think of the right hand side as defining a function f(), and of the equation as being simpl = f(). Eample 4: Find the domain of the following functions: f() = 3 g() = h() =

3 Eample 5: If f() = 6 + 4, write an epression for: [ ] [ ] f(1 + h) f(1) f(1 + h) + f(1). Eample 6: If P() = and we rewrite P() in the form what are the values of A and B? P() = A + B( 1) + C( 1)( 2) + D( 1)( 2)( 3), Eample 7: If we rewrite the function f() = 3 ( 1)( 2) in the form: what are the values of A, B, and C? f() = A + B 1 + C 2, Inverse of a function: Recall that two functions f() and g() are said to be inverse of each other if f(g()) = and g(f()) =. Eample 8: If h(t) = 3t + 7, find a function g(t) such that h(g(t)) = t. 3

4 Cartesian plane and the graph of a function: Points in a plane can be identified with ordered pairs of numbers to form the coordinate plane. To do this, we draw two perpendicular oriented lines (one horizontal and the other vertical) that intersect at 0 on each line. The horizontal line with positive direction to the right is called the -ais; the other line with positive direction upward is called the -ais. The point of intersection of the two aes is the origin O. The two aes divide the plane into four quadrants, labeled I, II, III, and IV. The coordinate plane is also called Cartesian plane in honor of the French mathematician/philosopher René Descartes ( ). An point P in the coordinate plane can be located b a unique ordered pair of numbers (a, b) as shown in the picture. The first number a is called the -coordinate of P; the second number b is called the -coordinate of P. II b O P(a,b) a I III IV Graphing functions: If f is a function with domain A, then the graph of f is the set of ordered pairs f(6) f() (6,f(6)) (,f()) graph of f = {(,f()) A}. f(2) (2,f(2)) In other words, the graph of f is the set of all points (,) such that = f(); that is, the graph of f is the graph of the equation = f() Obtaining information from the graph of a function: The values of a function are represented b the height of its graph above the -ais. So, we can read off the values of a function from its graph. In addition, the graph of a function helps us picture the domain and range of the function on the -ais and -ais as shown in the picture: 0 Range = f() Domain The graph of a function is a curve in the plane. But the question arises: Which curves in the -plane are graphs of functions? The vertical line test: A curve in the coordinate plane is the graph of a function if and onl if no vertical line intersects the curve more than once. Graph of a function Not a graph of a function 4

5 Lines and parabolas: The simplest tpes of functions are linear and quadratic functions. A linear function is a function f of the form f() = m + b, where m and b are real numbers. The graph of the equation = m + b is a (non-vertical) line in the -plane. The numbers m and b are called the slope and -intercept, respectivel. A quadratic function is a function f of the form f() = a 2 + b + c, where a,b, and c are real numbers and a 0. The graph of an quadratic function is a parabola; it can be obtained from the graph of f() = 2 b using shifting, reflecting and stretching transformations. Indeed, b completing the square a quadratic function f() = a 2 + b + c can be epressed in the standard form k Verte (h, k) (Minimum) k (Maimum) Verte (h, k) f() = a( h) 2 + k. h h The graph of f is a parabola with verte (h,k); the parabola opens upward if a > 0, or downward if a < 0. f() = a( h) 2 + k, a > 0 f() = a( h) 2 + k, a < 0 Eample 9: If the equation of the line through the points (3,4) and ( 1,6) is written as = A + B( + 1), what are the values of A and B? Eample 10: The parabola = intersects the -ais at the two points P and Q. What is the distance from P to Q? If we rewrite the inequalit < 0 in the form A < < B, what are the values of A and B? 5

6 Intersection points: The graphs of two equations intersect at a point if and onl if that point is a solution for both equations. Eample 11: Find the point(s) of intersection between the graph of the equation = 36 and the line with equation = 2; = 1. Eample 12: Find all points where the graph of ( 1) 2 2 = crosses the -ais; the -ais. Eample 13: The area of a right triangle is 7. The sum of the lengths of the two sides adjacent to the right angle of the triangle is 11. What is the length of the hpotenuse of the triangle? Eample 14: What is the smallest root of the polnomial Q() = ? 6

MA123, Chapter 1: Equations, functions, and graphs (pp. 1-15, Gootman)

MA123, Chapter 1: Equations, functions, and graphs (pp. 1-15, Gootman) MA123, Chapter 1: Equations, functions, and graphs (pp. 1-15, Gootman) Chapter Goals: Solve an equation for one variable in terms of another. What is a function? Find inverse functions. What is a graph?

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