4-3 Functions A relation is a function provided there is exactly one output for each input. It is NOT a function if one input has more than one output

Functions In order for a relationship to be a function… EVERY INPUT MUST HAVE AN OUTPUT TWO DIFFERENT INPUTS CAN HAVE THE SAME OUTPUT ONE INPUT CAN HAVE ONLY ONE OUTPUT INPUT Functions (DOMAIN) FUNCTION MACHINE OUTPUT (RANGE)

No two ordered pairs can have the same first coordinate Example 6 Which of the following relations are functions? R= {(9,10, (-5, -2), (2, -1), (3, -9)} S= {(6, a), (8, f), (6, b), (-2, p)} T= {(z, 7), (y, -5), (r, 7) (z, 0), (k, 0)} No two ordered pairs can have the same first coordinate (and different second coordinates).

Identify the Domain and Range. Then tell if the relation is a function. Input Output -3 3 1 1 3 -2 4 Function? Yes: each input is mapped onto exactly one output Domain = {-3, 1,3,4} Range = {3,1,-2}

Identify the Domain and Range. Then tell if the relation is a function. Input Output -3 3 1 -2 4 1 4 Domain = {-3, 1,4} Range = {3,-2,1,4} Notice the set notation!!! Function? No: input 1 is mapped onto Both -2 & 1

Is this a function? 1. {(2,5) , (3,8) , (4,6) , (7, 20)} 2. {(1,4) , (1,5) , (2,3) , (9, 28)} 3. {(1,0) , (4,0) , (9,0) , (21, 0)}

The Vertical Line Test If it is possible for a vertical line to intersect a graph at more than one point, then the graph is NOT the graph of a function. Page 117

Use the vertical line test to visually check if the relation is a function. (4,4) (-3,3) (1,1) (1,-2) Function? No, Two points are on The same vertical line.

Use the vertical line test to visually check if the relation is a function. (-3,3) (1,1) (3,1) (4,-2) Function? Yes, no two points are on the same vertical line

Examples I’m going to show you a series of graphs. **don’t write  Determine whether or not these graphs are functions. You do not need to draw the graphs in your notes. **or write this note

YES! Function? #1

#2 Function? YES! Y = 0.5x + 2 + 2sin(x) D: all reals R: all reals Another cool function: abs(x) + 2sin(x)

#3 Function? NO! Y = 0.5x + 2 + 2sin(x) D: all reals R: all reals Another cool function: abs(x) + 2sin(x)

#4 Function? YES! Y = 0.5x + 2 + 2sin(x) D: all reals R: all reals Another cool function: abs(x) + 2sin(x)

#5 Function? NO!

YES! Function? #6 This is a piecewise function

Function? #7 NO! D: all reals R: [0, 1] Another cool function: y = sin(abs(x)) Y = sin(x) * abs(x)

#8 Function? NO! Y = 0.5x + 2 + 2sin(x) D: all reals R: all reals Another cool function: abs(x) + 2sin(x)

YES! #9 Function?

Function Notation “f of x” Input = x Output = f(x) = y

(x, y) (x, f(x)) (input, output) y = 6 – 3x f(x) = 6 – 3x x y x f(x) Before… Now… y = 6 – 3x f(x) = 6 – 3x x y x f(x) -2 -1 1 2 12 -2 -1 1 2 12 (x, y) (x, f(x)) 9 9 6 6 3 3 (input, output)

Example. f(x) = 2x2 – 3 Find f(0), f(-3), f(5).