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Functions Lesson 1.3 The Magic Box • Consider a box that receives numbers in the top • And alters them somehow and sends a (usually) different number out the spout 14 3 -33 1 Today we look at the For each of the number mathematical way of combinations, can you figure talking about the out the "rule" which alters the "rule" number? What is a Function? • Definition: A function is a rule – Given X = { x1, x2, …} and Y = {y1, y2, …} – Assign to each element of X a unique element of Y • The set X is the domain – the set of all possible x values • The set Y is the range – the set of all resulting y values Notation • y = f(x) can be thought of as .. – “y is the image of the function f at x” or – “y is the value of the function f at the point x” • It is often read “y equals f of x” – your instructor will often use the phrase “f at x …” – this reflects the above Alternate Definition • A function is a set of ordered pairs: f(x) = { (x1, y1), (x2, y2), … } • Where no two ordered pairs have the same first element • Which of these is a function? { (3,4), (6,0), (9,4), (7,2)} {(1, 2), (3, 4), (5, 6), (1, 7)} {(8,-1), (9, -1), (10, -1), (105, -1)} Function Notation • Normal notation: f ( x) 2 x 7 x 2 • Functions can be defined/declared in your calculator: • Note the use of functional notation f(3) to evaluate functions. The -> is the STO> key Piecewise Functions • Some functions may be defined differently for different portions of the domain. x 1 if x 5 f ( x) x if x 5 • Your calculator can also define piecewise functions Piecewise Functions • Note the results when the function is graphed: – Actually there should be a gap between 6 and the square root of 5 – Which of the above values should apply for the function – Try the F3, trace on the graph Domain and Range • Either the domain or range or both can be restricted – due to the nature of the function • Consider f ( x) x 1 g ( x) x 2 5 x 6 x • Determine the domain and range for each: Domain and Range • f(x) 1 f ( x) x x g ( x) x 2 5 x 6 – Domain: all real numbers • BUT … not zero … why? – Range: y < -2 or y > 2 • g(x) – Domain: x < -3 or x > -2 (why?) – Range: y >= 0 Composition of Functions • Basic Concept: – value fed to first function – result fed to second function – end result taken from second function • Oft used notation: y = g(f(x)) or g ( f ( x)) g f ( x) Composition of Functions • Example – given : f ( x) 2 x 1 g ( x) 2 x • Try these f(4) = ? g(f(4)) = ? f(g(-2)) = ? Try defining the functions on your calculator and using the notation !! Assignment • Lesson 1.3A • Page 31 • Exercises: 3, 7, 11, 15, 21, 25, 27, 37, 43, 49 Defining a Graph • The graph of a function is: – the set of all points (x,y) which … – satisfy the function y = f(x) y x3 6 x 2 x 30 Some Graphs are NOT Functions • Which are not functions? Use the “vertical line” test. Intercepts • We are often concerned with where the graph intersects the axes – x-intercepts => when f(x) = y = 0 – y-intercepts => when x = 0, f(0) Reference Table of Functions Page 27, Text • Linear • Quadratics • Cubic • Absolute • Root functions • Reciprocals • Trig functions Symmetry of Graphs • Symmetric with the y-axis – f(-x) = f(x) for all x Called even functions Symmetry of Graphs • Symmetric about the origin – f(-x) = -f(x) Called odd functions Note: There are many functions which are neither odd nor even Transformation • Shift a function up or down y – k = f(x) – k > 0 => up – k < 0 => down • Shift function right or left y = f(x – h) – h > 0 => right – h < 0 => left Transformation • Define the function on the home screen • Enter different versions of f(x + h) Try for the following f(x)+k function: a*f(x) 3 f(-x) .1x .9 x f ( x) -f(x) Assignment • Lesson 1.3B • Pg 31 • Exercises 51, 53, 57, 59, 63, 67, 69