Read this section for an introduction to combinations of functions, then work through practice problems 1-9.
Some compositions are relatively common and easy, and you should recognize the effect of the composition on the graphs of the functions.
Example 3: Fig. 6 shows the graph of .
Graph (a) , (b) , and .
Solution: All of the new graphs are shown below in Fig. 7 .
(a) Adding 2 to all of the values of rigidly shifts the graph of 2 units upward.
(b) Multiplying all of the values of by 3 leaves all of the roots of fixed: if is a root of then and so is also a root of . If is not a root of , then the graph
of looks like the graph of stretched vertically by a factor of 3.
(c) The graph of is the graph of rigidly shifted 1 units to the right.
We could also get these results by examining the graph of , creating a table of values for and the new functions, and then graphing the new functions.
-1 | -1 | 1 | -3 | -2 | not definded |
0 | 0 | 2 | 0 | -1 | |
1 | 1 | 3 | 3 | 0 | |
2 | 1 | 3 | 3 | 1 | |
3 | 2 | 4 | 6 | 2 | |
4 | 0 |
2 | 0 | 3 | |
5 | -1 | 1 | -3 | 4 |
Practice 5: Fig. 8 is the graph of .
Graph (a) , (b) , (c) and (d) .