5.06  Functional Notation,

Functions, Domain, and Range

          Intermediate Algebra: One Step at a Time,  Pages  427 - 446:          

Dr. Robert J. Rapalje

Seminole State College of Florida

Sanford, FL  32773

Domain and Range (from a Graph)

Example 1:  Find the domain and range for the graph that is sketched below.  You may recognize that this is a parabola that opens downward with vertex at (0, −9). 

Window:                x=[10,10]  y=[-20,10]

Solution:  The sketch above has a vertex at  y=-9.  From this point, the graph extends all the way to the left and all the way to the right.  It also extends from y=-9, all the way down on the graph.  The graph never goes above y=-9.  Therefore, the domain is all real values.  The range is all values BELOW the vertex--that is,  or . 

Example 2:  Find the domain and range for the graph that is sketched below. 

Solution:     In this graph you may recognize the upper half of a parabola that extends upward and to the right to infinity.   The x-values of this graph extend from the vertex at x=-6, all the way to the right side of the graph and beyond.  Therefore, the domain  is .  The range is all values that are on or above the x-axis, or , or in interval notation . 

Example 3:  Find the domain and range for the graph that is sketched below. 

Solution:    You may recognize that this is the lower half of a parabola that extends downward and to the right to infinity.   The domain  is .   The range is all values below the x-axis, which can be written , or in interval notation . 

     In this graph you may recognize the lower half of a parabola that extends downward and to the right to infinity.   The x-values of this graph extend from the vertex at x=-6, all the way to the right side of the graph and beyond.  Therefore, the domain  is .  The range is all values that are on or above the x-axis, or , or in interval notation . 

Example 4:  Find the domain and range for the graph that is sketched below.

Solution:  From this graph, you may recognize this as the lower half of a circle that extends 3 units down, 3 units to the right, and 3 units to the left.  The domain  is .  The range is all values from -3 up to 0,  or in interval notation . 

Example 5:  Find the domain and range for the graph that is sketched below. 

Solution:  In the two sketches above, it is clear that there are points at (-1,0) and (6,0) that will be critical to finding the domain and range.   Notice that the graph actually touches the x axis at these two points, and from these points the graph extends upward from y=0.  The graph extends to the left from x=-1, and to the right from x=6.

From these graphs, it should be clear that the domain is .  Likewise, the range is all values that are on or above the x-axis, or , or in interval notation . 

Example 6:  Find the domain and range for the graph that is sketched below. 

Solution:     From this graph, it should be clear that the domain is .  Likewise, the range is all values that are on or above the x-axis, or , or in interval notation . 

Can you see that the graph extends in the x-direction from negative infinity to -4 and from 4 to infinity?  Therefore the domain is .  The values of y extend from negative infinity up to zero, so the range is   . 

Example 7:  Find the domain and range for the graph that is sketched below.

Solution:    You can see that this is the graph of a circle whose center is at the origin and whose radius is 4.  You can also see that the values of x extend from -4 to 4 inclusive, and the values of y also extend from -4 to 4 inclusive. 

Domain:    

Range:     

Example 8:  Find the domain and range for the graph that is sketched below.

Solution:    The domain consists of all values to left of and including -4 and to the right of and including 4  .  The range extends all the way down to negative infinity and all the way up to positive infinity.  This would be all real values or . 

Domain:     

Range:     

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