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Function

A FUNCTION is a mathematical “rule” that for each “input” (x-value) there is one and only one “output” (y – value). Set of Ordered Pairs : (input, output) or (x , y) No x-value is repeated!!! A function has a DOMAIN (input or x-values) and a RANGE (output or y-values)

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Function

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  1. A FUNCTION is a mathematical “rule” that for each “input” (x-value) there is one and only one“output” (y – value). Set of Ordered Pairs: (input, output) or (x , y) No x-value is repeated!!! A function has a DOMAIN (input or x-values) and a RANGE (output or y-values) For Graphs, Vertical Line Test: If a vertical line can be drawn anywhere on the graph that it touches two points, then the graph is not a function Function

  2. Function Representations:f is 2 times a number plus 5 Set of Ordered Pairs:{(-4, -3), (-2, 1), (0, 5), (1, 7), (2, 9)} Mapping: Table -4 -2 0 1 -2 Graph -3 1 5 7 9 Function Notation: f(-4) = -3 f(1) = 7 f(-2) = 1 f(2) = 9 f(0) = 5

  3. 8 - 4 2 4 -2 1 - 6 Examples of a Function #1: Graphs #4: Mapping #2: Table #3: Set { (2,3), (4,6), (7,8), (-1,2), (0,4), (-2, 5), (-3, -2)}

  4. 8 -4 2 -1 4 -2 1 Non – Examples of a Function #1: Graphs #2: Table #4: Mapping #3: Set {(-1,2), (1,3), (-3, -1), (1, 4), (-4, -2), (2, 0)}

  5. 1 -5 9 -2 0 -3 4 2 Practice: Is it a Function? #1: No • {(2,3), (-2,4), (3,5), (-1,-1), (2, -5)} • {(1,4), (-1,3), (5, 3), (-2,4), (3, 5)} • 3. 5. • 4. 6. #2: Yes #5: No #3: No #6: Yes #4: No

  6. Function Notation • Function Notation just lets us see what the “INPUT” value is for a function. (Substitution Statement) • It also names the function for us – most of the time we use f, g, or h. Examples: f(x) = 2x Reads as “f of x is 2 times x” f(3) = 2 * (3) = 6 The (3) replaces every x in rule for the input. Examples: g(x) = 3x2 – 7x Reads as “g of x is 3 times x squared minus 7 times x ” g(-1) = 3(-1)2 – 7(-1) = 10 The (-1) replaces every x in rule for the input.

  7. Given f: a number multiplied by 3 minus 5 f(x) = 3x – 5 1) Find f(-4) 2) Find f(2) 3) Find f(3x) = 3( -4) – 5 = -12– 5 = -17 = 3( 2) – 5 = 6 – 5 = 1 = 3( 3x) – 5 = 9x – 5 4) Find f(x + 2) 5) Find f(x) + f(2) = 3( x+ 2) – 5 = 3x + 6 – 5 = 3x + 1 = [3(x) – 5] + [3(2) – 5] = [3x – 5] + [1] = 3x – 4

  8. Given g: a number squared plus 6g(x) = x2 + 6 1) Find g(4) 2) Find g(-1) 3) Find g(2a) = ( 4)2 + 6 = 16 + 6 = 22 = ( -1)2 + 6 = 1 + 6 = 7 = ( 2a)2 + 6 = 4a2 + 6 5) Find g(x - 1) 4) Find 2g(a) = ( x-1)2 + 6 = x2 – 2x + 1 + 6 = x2 – 2x + 7 = 2[( a)2 + 6] = 2a2 + 12

  9. Operations on Functions Operations Notation: Sum: Difference: Product: Quotient: Example 1 Add / Subtract Functions a) b)

  10. Example 2 Multiply Functions a) b)

  11. Composite Function: Combining a function within another function. Notation: Function “f” of Function “g” of x “x to function g and then g(x) into function f” Example 1 Evaluate Composites of Functions Recall: (a + b)2 = a2 + 2ab + b2 a) b)

  12. Example 2 Composites of a Function Set a) g(x) f(x) f(g(x)) This means f(g(5))=3

  13. Example 2 Composites of a Function Set b) g(x) f(x) In set form, not every x-value of a composite function is defined

  14. Evaluate Composition Functions • Find: • f(g(3)) b) g(f(-1)) c) f(g(-4)) • d) e) f)

  15. Inverse Properties: 1] 2] Inverse Functions and Relations Inverse Relation: Relation (function) where you switch the domain and range values Function Inverse Function Inverse Domain of the function  Range of Inverse and Range of Function  Domain of Inverse Inverse Notation: Input a into function and output b, then inverse function will input b and output a (switch) Composition of function and inverse or vice versa will always equal x (original input)

  16. [3] Solve for y and replace it with Steps to Find Inverses [1] Replace f(x) with y [2] Interchange x and y One-to-One: A function whose inverse is also a function (horizontal line test) Function Inverse Inverse is not a function

  17. Example 1 One-to-One (Horizontal Line Test) Determine whether the functions are one-to-one. a) b) One-to-One Not One-to-One

  18. b) Example 2 Inverses of Ordered Pair Relations a) Are inverses f-1(x) or g-1(x) functions?

  19. Inverses of Graphed Relations FACT: The graphs of inverses are reflections about the line y = x Find inverse of y = 3x - 2 y = 1/3x + 2/3 x = 3y – 2 x + 2 = 3y 1/3x + 2/3 = y y= x y= 3x - 2

  20. Example 3 Find an Inverse Function a) b)

  21. Example 3 Continued c) d) PART D) Function is not a 1-1. (see example) So the inverse is 2 different functions: If you restrict the domain in the original function, then the inverse will become a function. (x> 0 or x < 0)

  22. Example 4: Verify two Functions are Inverses • Method 2: Composition Property • Method 1:Directly solve for inverse and check Yes, Inverses Yes, Inverses

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