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2.2 An Introduction to Functions

2.2 An Introduction to Functions. Definitions:. 1) Any set of ordered pairs ( x , y ) is called a relation. 2) A function is a relation such that for every first coordinate x in the ordered pair there is only one 2 nd coordinate y. i.e. No repetition in the first coordinate.

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2.2 An Introduction to Functions

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  1. 2.2 An Introduction to Functions

  2. Definitions: 1) Any set of ordered pairs (x, y) is called a relation 2) A functionis a relation such that for every first coordinate x in the ordered pair there is only one 2nd coordinate y. i.e. No repetition in the first coordinate.

  3. Ex1: Determine whether the relation represents y as a function of x. a) {(-2, 3), (0, 0), (2, 3), (4, -1)} Function b) {(-1, 1), (-1, -1), (0, 3), (2, 4)} Not a Function

  4. 3)The domain of the function is the set of all first coordinate of the ordered pairs. 4) The range of the function the set of all second coordinate of the ordered pair. Ex2: Consider the function {(0,0) , (-1,1), (2,3) } The domain is {0,-1,2} The range is {0,1,3}

  5. Function notation • Functions represented by equations are often named using a letter such as f, g, h, …. • To evaluate a function f (x) at x = a, substitute the specified value a for x into the given function.

  6. Ex3:

  7. one value of y function Solve for y two values of y not function How to identify if an equation represents a function or not? Case1:Algebraically Case 2:Graphically The vertical line test: A graph is the graph of a function if and only if no vertical line intersects the graph at more than one point.

  8. Ex4: Identify which one of the following relations define y as a function of x

  9. y y y y y x2 + y2 = 1 y = x2 y = x2 y2= x y2= x x x x x x Ex5: Identify which of the following graphs are graphs of functions b) c) a) function Not function Not function

  10. d) e) f) x = |y – 2| y=1 x=1 x function Not function Not function

  11. Def.: A one-to-one function is a function such that for every y, there is only onex that can be paired with y .[i.e. No repetition in either x or y] Ex6: Determine whether the relation represents y as a 1-1 function of x. a){(1,2), (2,2), (3,4)} b){(1,1), (1,3), (4,5)} c){(1,1), (2,2), (3,0)}

  12. y y 8 8 4 4 4 4 -4 -4 x x • HorizontalLine Test • A function y = f (x) is one-to-one if and only if no horizontal line intersects the graph of y = f (x) in more than one point. Ex7: Apply the horizontal line test to the graphs below to determine if the functions are one-to-one. b)y = x3 + 3x2– x – 1 a)y = x3 one-to-one not one-to-one

  13. Ex8: Find the domain of the following functions

  14. Definition of Domain • The domain of a function f is the set of all real numbers for which the function makes sense. How to find the domain?

  15. Example: Find Domain y x a c d b • Increasing, Decreasing, and Constant ●Increases on [c, d] ●Decreases on [a, b] ●Constanton [b, c] • Rules: • Start from left to right • Take the intervals from the x-axis

  16. Consider the graph of the function f(x) Ex9: y (-3, 6) [3, +∞). ●Increasing on (-∞, -3] x (3, -4) This graph is ●Decreasing on [-3, 3]

  17. EX10: • Piecewise-defined function

  18. The Greatest Integer Function ( Floor Function) Ex11: Find the value of

  19. Ex12 Let Find the value of f(-5)+f(5)

  20. Solve the following equations Ex13 Notes:

  21. Ex14:HW Find the domain of the function Ex15:HW Find the x-intercept and y-intercept of

  22. Q43/191 Sketch the graph of Ex16: Sketch the graph of The End

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