Read this section for an introduction to combinations of functions, then work through practice problems 1-9.
The absolute value function of a number , , is the distance between the number and . If is greater than or equal to , then is simply . If is negative, then which is positive since (negative number) = a positive number. On some calculators and in some computer programming languages, the absolute value
function is represented by .
Definition of : or .
The domain of consists of all real numbers. The range of consists of all numbers larger than or equal to zero, all non–negative numbers. The graph of
(Fig. 10) has no holes or breaks, but it does have a sharp corner at . The absolute value will be useful later for describing phenomena such as reflected light and bouncing balls which change direction abruptly or whose graphs
have corners.
The absolute value function has a number of properties which we will use later.
Properties of : For all real numbers and :
(a) . if and
only if .
(b)
(c)
Taking the absolute value of a function has an interesting effect on the graph of the function. Since
, then for any function we have
In other words, if , then so the graph of is the same as the graph of , then so the graph of is
just the graph of "flipped" about the x–axis, and it lies above the x–axis. The graph of will always be on or above the x–axis.
Example 5: Fig.
11 shows the graph of . Graph (a) , (b) and (c) .
Solution: The graphs are given in Fig. 12. In (b) we shift the graph of up 1 unit before taking the absolute value. In (c) we take the absolute value before shifting the
graph up 1 unit.
Practice 7: Fig. 13 shows the graph of . Graph (a) , (b) , and (c) .