El documento trata sobre las gráficas de funciones y sus propiedades, incluyendo la definición de una función y el test de la línea vertical. Se discuten funciones pares e impares, así como operaciones con funciones, y se presentan ejemplos de funciones constantes, lineales y cuadráticas, junto con sus gráficos. También se explican las funciones de valor absoluto y las funciones por tramos.

1. Graphs of Functions Recall: If f is a function from X to Y, then f = {(x,y) | y=f(x) and x in dom f } (x,y) in R 2 means (x,y) is a point on the Cartesian plane Def'n: The graph of the function f is the set of points in the Cartesian plane having coordinates ordered pairs in f .

2. Graphs of Functions Thus, we can find the range of a function f by graphing f .

3. Vertical Line Test A vertical line intersects the graph of a function in at most one point Examples: Which of the following is a function: 1. f = { (x,y) | 3x + y – 2 = 0 } 2. f (x) = x 2 3. {(x,y) | x 2 + y 2 = 1 }

4. Even and Odd Functions Def'n: Let f be a function. If for all x in dom f : 1. f (-x) = f (x) then f is an even function 2. f (-x) = - f (x) then f is an odd function Determine whether even or odd: 1. f (x) = x 4 + 2x 2 + 3 2. f (x) = x 3 + 3x

5. Odd and Even Functions Thm: Let f be a function. If f is 1. even then the graph of f is symmetric with respect to the y-axis 2. odd then the graph of f is symmetric with respect to the origin Example: 1. f (x) = x 2 2. f (x) = x 3

6. Function Notation Recall: y = f (x) if (x,y) is in f Given f (x) = 2x 2 + 3 and g ((x) = 3x 1/2 . Find: 1. f (0) 2. g (-4) 3. f (c + 1) 4. f (c) + f (1) 5. f ( g (1))

7. Operations on Functions Let f and g be functions. Then: 1. ( f ∓ g ) (x) = f (x) ∓ g (x) where dom( f ∓ g ) = dom f ∩ dom g 2. ( f g ) (x) = f (x) g (x) where dom( f g ) = dom f ∩ dom g 3. ( f/g ) (x) = f (x) / g (x) where dom( f/g ) = (dom f ∩ dom g ) \ {x | g (x) = 0}

8. Operations on Functions 4. ( f ∘ g ) (x) = f ( g (x)) where dom( f ∘ g ) = {x | x in dom g and g (x) in dom f } Given f (x) = 2x 2 + 3 and g ((x) = 3x 1/2 . Find fg , f-g, f/g, f ∘ g

9. Constant Functions functions of the form f (x) = c graph: horizontal line passing (0,c) dom f = R ran f = {c} ie. f (x) = 2

10. Linear Functions functions of the form f (x) = mx + b, m ≠ 0 graph: line with slope m and y-intercept b dom f = R ran f = R ie. f (x) = 2x + 3

11. Quadratic Functions functions of the form f (x) = a x 2 + bx + c, a ≠ 0 dom f = R ran f = see the graph (later) ie. f (x) = 2 x 2 + 3x + 4

12. Graphing Quadratic Equations Let f (x) = a x 2 + bx + c, a ≠ 0 if a < 0, parabola opens downward a > 0, parabola opens upward vertex of the parabola is at ( -b/2a, (4ac-b²)/4a) if b²-4ac is positive then the graph has two x-intercepts zero then the vertex lies on the x-axis negative then the graph has no x-intercepts

13. Absolute Value Functions Recall: f (x) = |x| = x if x is positve -x if x is negative To graph absolute value functions, we graph the expression inside the absolute value sign first and make the negative y-values positive. Then we adjust... (see demo)

14. Piecewise Functions functions defined differently for different intervals ie. f (x) = 2x if x < -1 x 2 if -1 ≤ x < 2 -1 if x =2 2x-6 if x > 2