Module 2: Lesson 2

 

Self-Check 3
  1. Questions 8.a. and 8.c. and question 11 on page 87


      1. The graph can be split up into four regions:

        • When x > 2, the square root graph will be undefined.
        • When 0 < x < 2, the y-value of the graph of the square root function will be greater than the y-value of the original graph.
        • When x < 0, the y-value of the graph of the square root function will be less than the y-value of the original graph.
        • When x = −2 and x = 0, the two graphs have the same y-value (they are invariant points).

        This should result in a graph similar to the following one:

         

        This illustration shows the graphs of two functions. One is the function y equals f brackets x brackets. The other one is the function y equals begin square root f brackets x brackets and the end of the square root. The two graphs have points of intersection at (2, 0) and (0, 1).


      1. In graphs of square root functions, the domains of the square root functions are restricted. Any portion of the original graph that lies below the x-axis will not appear in the square root graph. In this case, however, there are no circumstances where y < 0. There is a minimum on the range in the original function. Because this minimum is above the y-axis, the new minimum will be the square root of 1. As a result, note the graph of

         

        This illustration shows the graphs of two functions. One is a graph of the function y equals f brackets x brackets. The other is the graph of the function y equals begin square root f brackets x brackets end of the square root. There is a point of intersection at (0, 1).


      1. The graph can be split up into four regions:

        • When −0.41 < x < 2.41, the graph of the square root function will be undefined.
        • When −1 < x < −0.41 and when 2.41 < x < 3, the y-value of the square root graph will be greater than the y-value of the original graph.
        • When x < −1 and when x > 3, the y-value of the square root graph will be less than the y-value of the original graph.
        • When x = −1 and x = 3, the two graphs have the same y-value (they are invariant points).


        This should result in a graph that looks as follows:

         

        This illustration shows the graphs of two functions. One is a graph of the function y equals f brackets x brackets. The other graph is of the function y equals begin square root f brackets x brackets end of the square root. There are points of intersection at (-1, 1), (-0.41, 0), (2.41, 0), and (3, 1).

      2. Based on how previous questions have been solved, you should know that the graph of the original function will have the following domain and range:

         
        Domain: {x|x ∈ R}

        Range:

        		 {y|y ≥ −1, y ∈ R}

        In graphs of square root functions, the domains are restricted. Any portion of the original graph that lies below the x-axis will not appear in the square root graph. In this case, that portion occurs when −0.41 < x < 2.41. As a result, for

         
        Domain: {x|x ≤ −0.41, and x ≥ 2.41, x ∈ R}

        The minimum value of the original function is below the y-axis. This is the range of the square root function:

         

        Range:

        		 {y|y ≥ 0, y ∈ R}


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