5 1 review-offunctions

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5 1 review-offunctions

  1. 1. Review of Functions
  2. 2. <ul><li>Function – a rule which relates the values of one variable to the values of another variable </li></ul><ul><li>f , g , h – symbols for functions </li></ul><ul><li>y = f ( x ) – shows that f is the rule which assigns a unique value of y to a given value of x </li></ul><ul><li>x is the independent variable and y is the dependent variable </li></ul>
  3. 3. <ul><li>Domain – the set of input values for which f is defined </li></ul><ul><li>Range – the set of all values f assumes when x takes on values in the domain </li></ul><ul><li>Example. Consider f ( x ) = x 2 + 1 . </li></ul><ul><ul><li>The domain is the set of all real numbers. </li></ul></ul><ul><ul><li>The range is all real numbers f ( x ) ≥ 1 . </li></ul></ul>
  4. 4. <ul><li>Graph of a function f – the graph of the corresponding equation y = f ( x ) </li></ul><ul><li>sage: plot(x^3-7*x^2+x+6,(-2,6)) </li></ul>
  5. 5. <ul><li>Polynomial function: </li></ul><ul><li>Subtypes: quadratic ( f ( x ) = ax 2 + bx + c ), linear ( f ( x ) = ax + b ), constant ( f ( x ) = c ) </li></ul><ul><li>Examples: </li></ul>
  6. 6. <ul><li>Rational function: </li></ul><ul><li>Examples: </li></ul>
  7. 7. <ul><li>Composite function: </li></ul><ul><li>Example. f ( x ) = 2 x + 5 and g ( x ) = 3 x 2 </li></ul>
  8. 8. <ul><li>Piecewise-defined function – a function whose definition changes depending on the value of the independent variable </li></ul><ul><li>Example. </li></ul>
  9. 9. <ul><li>1.a Find the domain and range of y = x 2 – 9 . </li></ul><ul><li>The domain is the set of all real numbers. </li></ul><ul><li>The range is the set of all x such that x ≥ – 9 . </li></ul>
  10. 10. <ul><li>3. If f ( x ) = 4 x 2 + 7 , find . </li></ul>
  11. 11. <ul><li>7.a Graph the function </li></ul><ul><li>Connect (0, f (0)) and (1, f (1)) . </li></ul><ul><li>Then, connect (1, 2-1) and (2, f (2)) , with a “hole” at (1,1) . </li></ul>
  12. 12. <ul><li>8. Identify each function below as a polynomial , a rational function, a piecewise function, or none of these. </li></ul><ul><li>a) </li></ul><ul><li>b) . </li></ul><ul><li>c) . </li></ul><ul><li>d) f ( x ) = 5 </li></ul><ul><li>e) </li></ul>

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