Warm-Up  <ul><li>Write the equation, domain and range for each graph.  </li></ul>1. 2. 3.  f(x) = x 2  + 4x - 7,  find f(-...
Piecewise Functions Objectives:  Become familiar with piecewise functions Evaluate  piecewise functions
What Does Research Say? <ul><li>The function concept is one of the central concepts in all of mathematics   (Knuth, 2000; ...
Piecewise Functions A  piecewise function  is a function that is a combination of one or more functions.
Read this as “ f of x is 5 if x is greater than 0 and less than 13 ,  9 if x is greater than or equal to 13 and less than ...
Restricting the domain  of a function <ul><li>Use transformations to make a graph of  </li></ul><ul><li>What is the domain...
<ul><li>How could we define the domain?  </li></ul>Looking at only “part” or a “piece”  of the function  What rule would y...
Restricting the domain  of a function What is the domain?  All real numbers What is the equation for this graph?  f(x) = –...
<ul><li>How could we define the domain?  </li></ul>Looking at only “part” or a “piece” of the function  What rule would yo...
What rule would you write for this piecewise function? Piecewise Functions   x 2  – 3  if  x ≥ –2
a)  What is the value of y when  x = –4?  Give two ways to find it. Piecewise Functions   b) Which equation would you use ...
Piecing it all together:   Evaluating  Piecewise Functions <ul><li>Find the interval of the domain that contains the x-val...
2x + 1  if x ≤ 2  x 2  – 4  if x > 2  h(x) = Because  –1 ≤ 2, use the rule for x ≤ 2 . Because  4 > 2, use the rule for x ...
3x 2  + 1  if x < 0  5x – 2  if x ≥ 0  g(x) = Because  –1 < 0, use the rule for x < 0. Because  3 ≥ 0, use the rule for x ...
12  if x < –3  20  if x ≥ 6 f(x) = Because  –3 ≤  –1  < 6 , use the rule for – 3 ≤ x < 6   f(–1) = 15 Evaluate each piecew...
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Piecewise function lesson 3

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Piecewise function lesson 3

  1. 1. Warm-Up <ul><li>Write the equation, domain and range for each graph. </li></ul>1. 2. 3. f(x) = x 2 + 4x - 7, find f(-5).
  2. 2. Piecewise Functions Objectives: Become familiar with piecewise functions Evaluate piecewise functions
  3. 3. What Does Research Say? <ul><li>The function concept is one of the central concepts in all of mathematics (Knuth, 2000; Romberg, Carpernter, & Fennema, 1993; Yerushalmy & Schwartz, 1993). </li></ul><ul><li>Understanding multiple representations of functions and the ability to move between them is critical to mathematical development (Knuth, 2000; Rider, 2007). </li></ul>
  4. 4. Piecewise Functions A piecewise function is a function that is a combination of one or more functions.
  5. 5. Read this as “ f of x is 5 if x is greater than 0 and less than 13 , 9 if x is greater than or equal to 13 and less than 55 , and 6.5 if x is greater than or equal to 55 . ” The rule for a piecewise function is different for different parts, (or pieces), of the domain (x-values) For instance, movie ticket prices are often different for different age groups. So the function for movie ticket prices would assign a different value (ticket price) for each domain interval (age group).
  6. 6. Restricting the domain of a function <ul><li>Use transformations to make a graph of </li></ul><ul><li>What is the domain? </li></ul>all real numbers f(x) = x 2 - 3
  7. 7. <ul><li>How could we define the domain? </li></ul>Looking at only “part” or a “piece” of the function What rule would you write for this function? (How could we restrict the original function?) f(x) = x 2 - 3 if x ≥ -2 x ≥ -2 f(x) = x 2 - 3
  8. 8. Restricting the domain of a function What is the domain? All real numbers What is the equation for this graph? f(x) = –2x – 5
  9. 9. <ul><li>How could we define the domain? </li></ul>Looking at only “part” or a “piece” of the function What rule would you write for this function? f(x) = –2x – 5 f(x) = –2x – 5 if x < –2
  10. 10. What rule would you write for this piecewise function? Piecewise Functions x 2 – 3 if x ≥ –2
  11. 11. a) What is the value of y when x = –4? Give two ways to find it. Piecewise Functions b) Which equation would you use to find the value of y when x = 2? c) Which equation would you use to find the value of y when x = –2? x 2 – 3 if x ≥ –2
  12. 12. Piecing it all together: Evaluating Piecewise Functions <ul><li>Find the interval of the domain that contains the x-value </li></ul><ul><li>Then use the rule for that interval. </li></ul>9 3 -1 25
  13. 13. 2x + 1 if x ≤ 2 x 2 – 4 if x > 2 h(x) = Because –1 ≤ 2, use the rule for x ≤ 2 . Because 4 > 2, use the rule for x > 2. h(–1) = 2(–1) + 1 = –1 h(4) = 4 2 – 4 = 12 Evaluate the piecewise function for: x = –1 and x = 4.
  14. 14. 3x 2 + 1 if x < 0 5x – 2 if x ≥ 0 g(x) = Because –1 < 0, use the rule for x < 0. Because 3 ≥ 0, use the rule for x ≥ 0. g(3) = 5(3) – 2 = 13 g(–1) = 3(–1) 2 + 1 = 4 Evaluate each piecewise function for: x = –1 and x = 3
  15. 15. 12 if x < –3 20 if x ≥ 6 f(x) = Because –3 ≤ –1 < 6 , use the rule for – 3 ≤ x < 6 f(–1) = 15 Evaluate each piecewise function for: x = –1 and x = 3 15 if –3 ≤ x < 6 f(3) = 15 Because –3 ≤ 3 < 6 , use the rule for – 3 ≤ x < 6
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