If f(x) = 3x3 + 7 and g(x) = x2 + 55, then
The composition of functions is important. Consider two functions f and g defined by f(x) = x+6 and g(x) = x2. Then
We can also note the
If one thinks of f and g as rules, then an expression such as f(g(x)) is a rule that says "apply rule g, then apply rule f to what you get." Note that in this case, f(g(3)) is not the same as g(f(3)). The composition of functions lacks commutativity.
Here's a real-life application of one rule followed by another rule that will show that the order in which the rules are carried out is significant, since the results can differ depending on the order. Assume that you are facing north. Rule 1 says "take ten steps forward" and Rule 2 says "turn to your right." Imagine yourself doing Rule 1 followed by Rule 2. Now, go back to the starting point and do Rule 2 followed by Rule 1. Did you end up at the same spot as before?