Inequalties of Combined Functions Examples and Answers 8.4
Example 1 <ul><li>Let  and </li></ul><ul><li>Graph the functions on the same set of axes. Identify the points of intersect...
Solution <ul><li>The graph of  is a line with slope one, passing through the origin. The graph of  can be obtained by appl...
<ul><li>From the graph, the points of intersection appear to be (1,1) and (4,4). This can be verified algebraically by sol...
<ul><li>To verify that the points of intersection are (1,1) and (4,4), mentally substitute each of these ordered pairs int...
<ul><li>Method 2: Analyze the Difference Function </li></ul><ul><li>Another way to illustrate where one function is greate...
<ul><li>This function is positive on the interval (1,4). Therefore, </li></ul><ul><li>on the interval (1,4) </li></ul><ul>...
<ul><li>To make this more clearly visible, graph the combined function using a graphing calculator with an appropriate vie...
<ul><li>It appears that  on the interval (1,4). However,  </li></ul><ul><li>is not defined for  .Determine    and    to de...
Example 2 <ul><li>Solve </li></ul><ul><li>This is a problem that cannot be solved by using algebra; it must be solved usin...
Solution <ul><li>graph  and  </li></ul><ul><li>make sure the calculator is in Radian mode </li></ul><ul><li>change the win...
8.4
<ul><li>the graph of  is above the graph of  </li></ul><ul><li>between    and   , and between    and  </li></ul><ul><li>th...
Example 3 <ul><li>A computer store’s cost, C, for shipping and storing n computers can be modeled by the function  </li></...
Solution <ul><li>Use a graphing calculator. Apply number sense to choose appropriate window settings. </li></ul><ul><li>Th...
<ul><li>Method 1: Graphing Calculator </li></ul><ul><li>Graph the adjusted cost function  </li></ul><ul><li>and identify t...
<ul><li>Notice that the adjustment term translates the original cost curve down 1500 units, making it easy to distinguish ...
<ul><li>According to the graph, an inventory order of form 169 to 841 units will keep the costs below $1500. This region c...
<ul><li>Method 2: Use Pencil and Paper </li></ul><ul><li>The   quadratic expression on the left side corresponds to a quad...
<ul><li>Since the warehouse has a maximum storage capacity of 750 units, a shipment of from 159 to 750 computers will keep...
<ul><li>The minimum cost occurs when approximately 356 computers are ordered. The corresponding cost for this size of orde...
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Inequalties Of Combined Functions2[1]

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Inequalties Of Combined Functions2[1]

  1. 1. Inequalties of Combined Functions Examples and Answers 8.4
  2. 2. Example 1 <ul><li>Let and </li></ul><ul><li>Graph the functions on the same set of axes. Identify the points of intersection. </li></ul><ul><li>Illustrate the regio0n for which </li></ul><ul><li>i) ii) </li></ul>8.4
  3. 3. Solution <ul><li>The graph of is a line with slope one, passing through the origin. The graph of can be obtained by applying a horizontal translation of 2 units to the right of the graph </li></ul>8.4
  4. 4. <ul><li>From the graph, the points of intersection appear to be (1,1) and (4,4). This can be verified algebraically by solving the linear-quadratic system of f(x) and g(x). Set the two functions equal and solve for x. </li></ul>8.4
  5. 5. <ul><li>To verify that the points of intersection are (1,1) and (4,4), mentally substitute each of these ordered pairs into and </li></ul><ul><li>Method 1: Compare the Functions Visually </li></ul><ul><li>Using different colours or line styles is a useful tequnieq for visulaizing regions for which one function is greater than another. The graph of y= f(x) is above the graph of y= g(x) between x =1 and x =4. Therfore, </li></ul><ul><li>f(x) >g(x) on the interval (1,4) </li></ul><ul><li>g(x) >f(x) on the interval (- ∞,1) U (4,∞) </li></ul>8.4
  6. 6. <ul><li>Method 2: Analyze the Difference Function </li></ul><ul><li>Another way to illustrate where one function is greater then another is to subtract the functions and see where the graph of difference is above the x-axis. </li></ul><ul><li>The function, ,is positive on the intervals where and therefore </li></ul>8.4
  7. 7. <ul><li>This function is positive on the interval (1,4). Therefore, </li></ul><ul><li>on the interval (1,4) </li></ul><ul><li>on the intervals (-,1) U (4,) </li></ul><ul><li>Method 3: Analyse the Quotient Function </li></ul><ul><li>The function f(x) and g(x) can also be compared by analyzing their quotient. Graph the combined function and identify the intervals for which </li></ul><ul><li>this quotient is greater than one, which will correspond to where </li></ul>8.4
  8. 8. <ul><li>To make this more clearly visible, graph the combined function using a graphing calculator with an appropriate viewing window. Include a graph of y=1. Use the Intersect operation to identify the coordinates of the points of intersection. </li></ul>8.4
  9. 9. <ul><li>It appears that on the interval (1,4). However, </li></ul><ul><li>is not defined for .Determine and to decide what happens when </li></ul><ul><li>Thus, and </li></ul><ul><li>i) on the interval (1,4) </li></ul><ul><li>ii) on the intervals (-,1) U (4, ) </li></ul>8.4
  10. 10. Example 2 <ul><li>Solve </li></ul><ul><li>This is a problem that cannot be solved by using algebra; it must be solved using a graphing calculator in a few simple steps: </li></ul>8.4
  11. 11. Solution <ul><li>graph and </li></ul><ul><li>make sure the calculator is in Radian mode </li></ul><ul><li>change the window settings </li></ul><ul><li>use the Intersection operation to find the x-coordinates </li></ul><ul><li>of any points of intersection </li></ul><ul><ul><li>there is an asymptote at </li></ul></ul><ul><ul><li>there are points of intersection at (2.70, 0.43), (7.33, 0.87), (8.26, 0.92) </li></ul></ul><ul><li>  </li></ul><ul><li>  </li></ul><ul><li>  </li></ul>8.4
  12. 12. 8.4
  13. 13. <ul><li>the graph of is above the graph of </li></ul><ul><li>between and , and between and </li></ul><ul><li>therefore: (0, 2.70) U (7.33, 8.26) </li></ul><ul><li>- this can be verified by using the TEST in the calculator menu </li></ul><ul><li>- enter the inequality in the Y= </li></ul><ul><li>- set calculator to Dot mode </li></ul><ul><li>- where the equation is true, a line will appear on the graph </li></ul>8.4
  14. 14. Example 3 <ul><li>A computer store’s cost, C, for shipping and storing n computers can be modeled by the function </li></ul><ul><li>The storage capacity of the store’s warehouse is 750 units. </li></ul><ul><li>Graph this function and explain its shape. What is the domian of interest for this problem? </li></ul><ul><li>Determine the maximum and minimum number of computers that can be ordered at any time to keep costs below $1500, assuming that inventory has fallen below zero. </li></ul><ul><li>What is the optimum order size that will minimize storage costs? </li></ul><ul><li>Why might this not be the best number to order? </li></ul>8.4
  15. 15. Solution <ul><li>Use a graphing calculator. Apply number sense to choose appropriate window settings. </li></ul><ul><li>The shape of this graph can be understood if it is thought of as the superposition of a linear function and a rational function. </li></ul><ul><li>The domain of is { n ER, n >0} </li></ul><ul><li>b) Determine the values of n that will keep costs below $1500. </li></ul>8.4
  16. 16. <ul><li>Method 1: Graphing Calculator </li></ul><ul><li>Graph the adjusted cost function </li></ul><ul><li>and identify the region below the horizontal axis. Apply number sense to choose the region below the horizontal axis. Apply number sense to choose appropriate window settings to view the region of interest. </li></ul>8.4
  17. 17. <ul><li>Notice that the adjustment term translates the original cost curve down 1500 units, making it easy to distinguish the region of interest. Locate the zeros by using the Zero operation. </li></ul>8.4
  18. 18. <ul><li>According to the graph, an inventory order of form 169 to 841 units will keep the costs below $1500. This region can be illustrated by setting Y2 =0 and then shading above and below for the line styles of Y1 and Y2 , respectfully. </li></ul><ul><li>The region where the two shaded parts overlap shows the interval for which costs are below $1500. </li></ul>8.4
  19. 19. <ul><li>Method 2: Use Pencil and Paper </li></ul><ul><li>The quadratic expression on the left side corresponds to a quadratic function that opens upward. The region between the zeros will correspond to n-values for which . Solve for these zeros. </li></ul>8.4
  20. 20. <ul><li>Since the warehouse has a maximum storage capacity of 750 units, a shipment of from 159 to 750 computers will keep costs below $1500. </li></ul><ul><li>c) To determine the optimum order size that will minimize shipping and storage costs, locate the minimum of the original cost function, </li></ul><ul><li>Use the Minimum operation of the graphing calculator. </li></ul>8.4
  21. 21. <ul><li>The minimum cost occurs when approximately 356 computers are ordered. The corresponding cost for this size of order is approximately $1095. </li></ul><ul><li>d) The optimum order size may not be desirable, particularly if consumer demand is not aligned with this value. If demand significantly exceeds 365 units, then profit will be lost due to unfilled sales requests. On the other hand, if demand is significantly below 365 units, unnecessary shipping and store costs will occur, as well as the cost of inventory space for other products. </li></ul>8.4
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