6.2: Operations on functions given by tables
( \newcommand{\kernel}{\mathrm{null}\,}\)
We can also combine functions that are defined using tables.
Let f and g be the functions defined by the following table.
x1234567f(x)6314076g(x)4025−231
Describe the following functions via a table:
- 2⋅f(x)+3
- f(x)−g(x)
- f(x+2)
- g(−x)
Solution
For (a) and (b), we obtain by immediate calculation
x12345672⋅f(x)+315951131715f(x)−g(x)23−1−1245
For example, for x=3, we obtain 2⋅f(x)+3=2⋅f(3)+3=2⋅1+3=5 and f(x)−g(x)=f(3)−g(3)=1−2=−1.
For part (c), we have a similar calculation of f(x+2). For example, for x=1, we get f(1+2)=f(1+2)=f(3)=1.
x1234567−10f(x+2)14076 undef. undef. 63
Note that for the last two inputs x=6 and x=7 the expression f(x+2) is undefined, since, for example for x=6, it is f(x+2)=f(6+2)=f(8) which is undefined. However, for x=−1, we obtain f(x+2)=f(−1+2)=f(1)=6. If we define h(x)=f(x+2), then the domain of h is therefore Dh={−1,0,1,2,3,4,5}.
Finally, for part (d), we need to take x as inputs, for which g(−x) is defined via the table for g. We obtain the following answer.
x−1−2−3−4−5−6−7g(−x)4025−231
Let f and g be the functions defined by the following table.
x1357911f(x)3511497g(x)7−691195
Describe the following functions via a table:
- f∘g
- g∘f
- f∘f
- g∘g
Solution
The compositions are calculated by repeated evaluation. For example,
(f∘g)(1)=f(g(1))=f(7)=4
The complete answer is displayed below.
x1357911(f∘g)(x)4 undef. 97911(g∘f)(x)−695 undef. 911(f∘f)(x)5117 undef. 94(g∘g)(x)11 undef. 9599