Each element in A must be matched with an element in B Ex– (0,3) (3,2) (9,4) (12,5) Some elements in the range may not be matched with the domain. Two or more elements in the domain may be matched with one element in the range. An element in the domain CANNOT be matched with two different elements in the range.

 Example— Is the following relationship a function? DomainRange

Example— A = {a, b, c}B = {0, 1, 2, 3} Which sets of ordered pairs represent functions from A to B? {(a, 1), (b, 2), (c, 3)} {(1, a), (0, a), (2, c), (3, b) }

 Which of the equations represent(s) y as a function of x? x 2 + y = 1 - x + y 2 = 1

 Vertical Line Test!!!

 f(x) reads “f of x”  Evaluating functions Example— G(x) = -x 2 + 4x + 1 G(2) = G(t) = G(x + 2) =

 Domain – Input values these are the x values in an equation  Range – Output values these are the y values in an equation  In an equation– you must exclude any of the values that make the denominator zero or make a negative under the radical.

Example If f(x) = x 2 + 3, evaluate f(2) 2 2 f( ) = 2 + 3x x f(2) = f(2) = 7 What does this mean? It means when x = 2, y = 7 It means the point (2,7) is on the graph

Example If f(x) = 2x 3 + 4x - 6, evaluate f(-1) (-1) f( ) = x x f(-1) = 2(-1) + 4(-1) - 6 f(-1) = -12 What does this mean? It means when x = -1, y = -12 It means the point (-1,-12) is on the graph of f(x) x (-1)

The evaluation may get slightly more complicated… Example If g(x) = x 2 + 2x, evaluate g(x – 3) g(x) = x x (x -3) g(x) = (x 2 – 6x + 9) + 2x - 6 g(x) = x 2 – 6x x - 6 g(x) = x 2 – 4x + 3 What does this mean?

Definition: Piecewise Function –a function defined by two or more functions over a specified domain.

What do they look like? f(x) = x 2 + 1, x  0 x – 1, x  0 You can EVALUATE piecewise functions. You can GRAPH piecewise functions.

Evaluating Piecewise Functions: Evaluating piecewise functions is just like evaluating functions that you are already familiar with. f(x) = x 2 + 1, x  0 x – 1, x  0 Let’s calculate f(2). You are being asked to find y when x = 2. Since 2 is  0, you will only substitute into the second part of the function. f(2) = 2 – 1 = 1

f(x) = x 2 + 1, x  0 x – 1, x  0 Let’s calculate f(-2). You are being asked to find y when x = -2. Since -2 is  0, you will only substitute into the first part of the function. f(-2) = (-2) = 5

Your turn: f(x) = 2x + 1, x  0 2x + 2, x  0 Evaluate the following: f(-2) =-3 ? f(0) =2 ? f(5) =12 ? f(1) =4 ?

One more: f(x) = 3x - 2, x  -2 -x, -2  x  1 x 2 – 7x, x  1 Evaluate the following: f(-2) =2 ? f(-4) =-14 ? f(3) =-12 ? f(1) =-6 ?

Graphing Piecewise Functions: f(x) = x 2 + 1, x  0 x – 1, x  0 Determine the shapes of the graphs. Parabola and Line Determine the boundaries of each graph. Graph the parabola where x is less than zero.       Graph the line where x is greater than or equal to zero.       

      3x + 2, x  -2 -x, -2  x  1 x 2 – 2, x  1 f(x) = Graphing Piecewise Functions: Determine the shapes of the graphs. Line, Line, Parabola Determine the boundaries of each graph.             