4.3C   Piecewise Functions Analysis

Piecewise Functions Analysis

The piecewise function at the right has 3 equations with their conditions. 

     1.  y = -x + 1, if the domain is between -3 and 1

     2.  y = 2, if the domain is 1 

     3.  y = x2, if the domain is greater than 1

   Looking at each function separately, and using the given conditions for the domain, x, we will be able to determine if this is a function and will be able to sketch a graph of the function.

     1.  y = -x + 1

               ~  y = -x + 1 is a linear equation with a negative slope.

               ~  Think about where it will start and where it will end.  Look at the condition [-3, 1).

               ~  -3 is included, so f(-3) = 4 will be an endpoint.

               ~  1 is not included, so f(1) = 0 will be a hole.

               ~  Sketch this linear equation from the endpoint [-3, 4) to the hole (1, 0).

     2.  y = 2

             ~  y = 2 is a constant function.

             ~  Look at the condition:  x = 1, so f(1) = 2.

             ~  This is a point (1, 2).

             ~  Plot a point at (1, 2)

    3.  y = x2

           ~  y = x2, is a quadratic equation.

           ~  Look at the condition:  x > 1.  x doesn't equal 1, but must be greater than 1.

           ~  f(2) = 4, f(3) = 9...

           ~ The graph will be a parabola greater than 1 to infinity.

           ~  Sketch this parabola from 1 using the points we generated.  

           ~  At x = 1, draw a hole because the point does not exist, but all points from .000001 to infinity exist.

    4.   Determine the domain of f(x).

          ~ To find the domain of f(x), look at its definition.

          ~  Since f is defined for all x greater than or equal to 2, the domain is x ≥ -3.

          ~  Using interval notation [-3, ∞)

    5.  Determine the range of f(x).

         ~  Look at the graph to determine the range.

         ~  The range will be y > 0.

         ~  Using interval notation (0, ∞).

    6.  Continuity

         ~  Remember, a function is continuous if we don't have jumps, holes, or gaps in the function.

        ~  If we can draw a graph without lifting the pencil from the paper, it is said to be continuous.

        ~  If it's not continuous, we say that the function is discontinuous.

         ~  Looking over your graph you should determine that this function is discontinuous.