d The domain is the possible x-values and the range is the possible y-values of the relations. They can be determined by looking at the graphs.
A
aDiagram:
B
bDiagram:
C
c See solution.
D
dDomain of y=|x|: All real values of x.
Range of y=|x|: y≥0 Domain of |y|=x: x≥0 Range of |y|=x: All real values of y.
Practice makes perfect
a This is an absolute value function. To understand how to graph it, we should start with y=x. The graph of this function is a straight line with a slope of 1 that goes through the origin
When we take the absolute value of x, all negative outputs becomes positive. This means all points with negative y-values gets reflected in the x-axis.
b Now we take the absolute value of y which means we cannot substitute negative x in the function. To determine a few points that fall on this graph, we will substitute different values of y and evaluate the corresponding value of x.
|c|c|c|
[-1em]
y & |y| & x [0.2em]
[-1em]
-3 & | -3| & 3 [0.2em]
[-1em]
-2 & | -2| & 2 [0.2em]
[-1em]
-1 & | -1| & 1 [0.2em]
[-1em]
0 & | 0| & 0 [0.2em]
[-1em]
1 & | 1| & 1 [0.2em]
[-1em]
2 & | 2| & 2 [0.2em]
[-1em]
3 & | 3| & 3 [0.2em]
Now we can plot the relation by marking the points in a coordinate plane and drawing a graph through them.
c To examine the similarities and differences, let's draw the two graphs in the same coordinate plane.
The graphs are identical in the coordinate plane's first quadrant. This is the only similarity. The difference is that y=|x| cannot assume negative outputs while |y|=x cannot assume negative inputs.
d The domain and range of a function describes the possible x- and y-values of the function. They can be determined by examining their diagrams.
Domain
The graph of y=|x|, enters the coordinate plane from the left and exits on the right. Since there are no restrictions in what inputs can be used, the domain must be all real values of x.
y=|x|:& All real values of x
Regarding |y|=x, we see that it has its vertex at the origin but it does not cross the y-axis Therefore, its domain is all values of x greater than or equal to 0.
|y|=x:& x≥ 0
Range
When it comes to the range, we notice that y=|x| has a vertex in the origin but it does not cross the x-axis. Therefore, its range is all values of y greater than or equal to 0.
y=|x|:& y≥ 0
As for |y|=x, we see that it comes in from below and exits on the top. Since there are no restrictions in what outputs can be used, its range is all real values of y.
|y|=x:& All real values of y
